# Full text of "The Development Of Mathematics Second Edition"

## See other formats

The Development of Mathematics THE DEVELOPMENT of MATHEMATICS BY E. T. Bell Professor of Mathematics, California Institute of Technology Second Edition McGRAW-HILL BOOK COMPANY, Inc. York • J^ondon 1945 THE DEVELOPMENT OF MATHEMATICS Copyright, 1940, 1945, by E. T. Bell PRINTED IN THE UNITED STATES OF AMERICA All rights reserved. This book, or farts thereof, may not be reproduced in any form without permission of the author. IX 04330 To Any Prospective Reader Nearly fifty years ago an American critic, reviewing the first volume (1888) of Lie’s Theorie der Transformationsgruppen, set his own pace (and ours) in the following remarks. There is probably no other science which presents such different appear- ances to one who cultivates it and one who does not, as mathematics. To [the noncultivator] it is ancient, venerable, and complete; a body of dry, irrefutable, unambiguous reasoning. To the mathematician, on the other hand, his science is yet in the purple bloom of vigorous youth, everywhere stretching out after the "attainable but unattained,” and full of the excitement of nascent thoughts; its logic is beset with ambiguities, and its analytic processes, like Bunyan’s road, have a quagmire on one side and a deep ditch on the other, and branch off into innumerable by-paths that end in a wilderness.* Once we venture beyond the rudiments, we may agree that those who cultivate mathematics have more interesting things to say than those who merely venerate. Accordingly, we shall follow the cultivators in their explorations of a Bunyan’s road through the development of mathematics. If occasionally we have no eyes for the purple bloom, it trill be because we shall need all our faculties to avoid falling into the ditch or wandering off into a wilderness of trivialities that might be mistaken for mathematics or for its history. And we shall leave to antiquarians the difficult and delicate task of restoring the roses to the cheeks of mathe- matical mummies. The course chosen in the following chapters was determined by two factors. The first was the request from numerous corre- spondents, principally students and instructors, for a broad account of the general development of mathematics, with par- ’ C. H. Chapman, Bulletin cf the Nets York Mathematical Society, 2, 1S92, 61. VI TO ANY PROSPECTIVE READER ticular reference to the main concepts and methods that have, in some measure, survived. The second was personal association for several years with creative mathematicians in both the pure and the applied divisions. Not a history of the traditional kind, but a narrative of the decisive epochs in the development of mathematics was wanted. A large majority asked for technical hints, where possible with- out too great detail, why certain things continue to interest mathematicians, technologists, and scientists, while others are ignored or dismissed as being no longer vital. Many who planned to end their mathematical education with the calculus, or even in some instances earlier, wished to be shown something of the general development of mathematics beyond that outstanding landmark of seventeenth-century thought, as part of a civilized education. Those intending to continue in mathematics or science or technology also asked for a broad general treatment with technical hints. They gave two additional reasons, the second of singular interest to any professed teacher. They be- lieved that a survey of the main directions along which living mathematics has developed would enable them to decide more intelligently in what particular field of mathematics, if any, they might find a lasting satisfaction. The second reason for their request was characteristic of a generation that has grown rather tired of being told what to think and whom to respect. These candid young critics of their would-be educators hoped that a cursory personal inspection of the land promised them, even from afar off, would enable them to resist the blandishments of persuasive ‘subdividers’ bent on selling their own tracts to the inexperienced. We seem to have come a long way since 1873, when that erudite English historian of mathematics and indefatigable manufacturer of drier-than- dust college textbooks, Isaac Todhunter (1820-1884), counseled a meek docility, sustained by an avid credulity, as the path of intellectual rectitude: If he [a student of mathematics] does not believe the statements of his tutor, probably [in Todhunter’s day at Cambridge] a clergyman of mature knowledge, recognized ability and blameless character — his suspicion is irrational, and manifests a want of the power of appreciating evidence, a want fatal to his success in that branch of science which he is supposed to be cultivating. Be the wisdom of Todhunter’s admonition what it may, it is astonishing how few students entering serious work in mathe- TO ANY PROSPECTIVE READER vii matics or its applications have even the vaguest idea of the highways, the pitfalls, and the blind alleys ahead of them. Consequently, it is the easiest thing in the world for an enthusi- astic teacher, “of mature knowledge, recognized ability and blameless character,” to sell his misguided pupils a subject that has been dead for forty or a hundred years, under the sincere delusion that he is disciplining their minds. With only the briefest glimpse of what mathematics in this twentieth century — not in 2100 b.c. — is about, any student of normal intelligence should be able to distinguish between live teaching and dead mathematics. He will then be less likely than his confiding com- panion to drown in the ditch or perish in the wilderness. Many asked for some reference to the social implications of mathematics. A classic strategy in mathematics is the reduction of an unsolved problem to one already solved. It seems plausible that more than half the problem of mathematics and society is reducible to that of the physical sciences and society. There being as yet no widely accepted solution of the latter problem, we shall leave the former with the reduction indicated. Anyone will thus be able to reach his own conclusions from that solution of the scientific problem which he accepts. Proposed solutions range from Platonic realism at one extreme to Marxian deter- minism at the other. Occasional remarks may suggest an inquiry into the equally difficult question of w r hat part civilization, with its neuroses, its wars, and its national jealousies, has played in mathematics. These asides may be of interest to those intending to make mathematics their lifework. Incidentally, in this con- nection, I was told that I might write for adults. Chronological age is not necessarily a measure of adulthood; a first-year student in a university may be less infantile, in everything but mathe- matics, than the distinguished savant lecturing at him. The topics selected for description were chosen after con- sultation with numerous professionals who know from hard personal experience %vhat mathematical invention means. On their advice, only main trends of the past six thousand years are considered, and these are presented only through typical major episodes in each. As might be anticipated by any tvorker in mathematics, the conclusions reached by following such advice differ occasionally from those hallowed by the purely historical tradition. Wherever this is so, references to other accounts will enable any reader to form his own opinion. There are no abso- lutes (except possibly this) in mathematics or in its history. vm TO ANY PROSPECTIVE READER Most of the differences reflect two possible and sometimes divergent readings of mathematical evolution. Whoever has him- self attempted to advance mathematics is inclined to be more skeptical than the average spectator toward any alleged antici- pation of notable progress. From his own experience and that of others still living, the professional mathematician suspects that often what looks like an anticipation after the advance was made was not even aimed in the right direction. From many a current instance, he knows further that when at length progress started, it proceeded along lines totally different from those which, in retrospect, it ‘should 5 have followed. Nothing is easier, on the other hand, than to fit a deceptively smooth curve to the discontinuities of mathematical invention. Everything then appears as an orderly progression from the Egypt of 4000 b.c. and the Babylon of 2000 b.c. to the Gottingen of 1934 and the U.S.A. of 1945, with Cavalieri, for instance, indistinguishable from Newton in the neighborhood of the calculus, or Lagrange from Fourier in that of trigonometric series, or Bhaskara from Lagrange in the region of Fermat’s equation. Professional historians may sometimes be inclined to overemphasize the smoothness of the curve; professional mathe- maticians, mindful of the dominant part played in geometry by the singularities of curves, attend to the discontinuities. This is the origin of most differences of opinion between the majority of those who cultivate mathematics and the majority of those who do not. That such differences should exist is no disaster. Dissent is good for the souls of all concerned. No apology need be tendered the thousands of dead and living mathematicians whose names are not mentioned. Only a meaningless catalogue could have cited a tenth of those who have created mathematics. Nor, when between 4,000 and 5,000 papers and books devoted to mathematical research — the cre- ation of new mathematics — are being published every 365 days, is there any point in attempting to minimize the omission of certain topics that have interested, and may still interest, hundreds of these unnamed thousands. However, what a suffi- cient number of competent men consider the vital things are at least mentioned. Anyone desirous of following the detailed his- tory of certain major developments will find the technical histories of special topics, written by mathematicians formathe- maticians, ample for a beginning. Some of these severely technical histories extend to hundreds of pages, a few to thou- TO ANY PROSPECTIVE READER ix / sands; they refer to the labors of thousands of men, most of whom are all but completely forgotten. Yet, like the tiny crea- tures whose empty frames survive in massive coral reefs that can wreck a battleship, these hordes of all but anonymous mathe- maticians have left something in the structure of mathematics more durable than their own brief and commonplace lives. As to the mechanical features of the book, the inevitable foot- notes have been kept to a minimum by the simple expedient of throwing hundreds away. Some direct those seeking further information on the relevant mathematics to works by creative mathematicians. Other things being equal, preference is given works containing extensive bibliographies compiled by experts hairing firsthand knowledge of the subjects treated. A superscript number indicates a footnote; all are collected for easy reference just before the index. All should be ignored till a possible return to some point. The index will be found helpful. Men’s initials and dates (except a very few, unobtainable without undue labor), seldom repeated in the text, are given in the index; cross references to definitions, etc., are avoided by the same means. Nationalities arc stated; if more than one country has a claim to some man, the place w'here he did most of his work is given. On a previous occasion {Men of mathematics, Simon & Schuster, New York, 1937), I almost precipitated an inter- national incident by calling a Pole a Russian. I trust that few such disastrous blunders will be found here. The book mentioned contains full-length biographies of about thirty-five leading mathematicians of the past. Dates in the text appended to mathematical events serve two purposes, the first of which is obvious. The second is to avoid elaborate references. The date, if later than 1636 and earlier than 1868, will usually enable anyone seriously interested to locate the matter concerned in the collected works of the author cited; if later than 1867, and whether or not collected works arc available, the exact reference, with a concise abstract of the work, is given in the annual Jahrbuch ilbcr die Fortschrittc der Mathematik. For the period beginning in 1931, the Zcntralblatt fur Mathematik und Hire Grenzgebiete serves the same purpose. The American Mathematical Reviews, 1940-, is of the same general character as the German abstract journals. Compara- tively scarce early periodicals, likely tobc found only in specialized libraries, are not cited, although they were frequently con- X TO ANY PROSPECTIVE READER suited. This omission may be partly compensated by refer- ring to the German and French mathematical encyclopedias listed in the notes. Other references to sources before 1637 are given in the proper places. For the period before 1637, the works of professional his- torians of mathematics have been used for some matters on which the historians arc in approximate agreement among them- selves. Theirs is a difficult and exacting pursuit; and if con- troversies over the trivia of mathematics, of but slight interest to either students or professionals, absorb a considerable part of their energies, the residue of apparently sound facts no doubt justifies the inordinate expense of obtaining it. Without the devoted labors of these scholars, mathematicians would know next to nothing, and perhaps care less, about the first faltering steps of their science. Indeed, an eminent French analyst of the twentieth century declared that neither he nor any but one or two of his fellow professionals had the slightest interest in the history of mathematics as conceived by historians. He amplified his statement by observing that the only history of mathematics that means anything to a mathematician is the thousands of technical papers cramming the journals devoted exclusively to mathematical research. These, he averred, are the true history of mathematics, and the only one either possible or profitable to write. Fortunately, I am not attempting to write a history of mathematics; I hope only to encourage some to go on, and decide for themselves whether the French analyst was right. Preference has been given in citing purely historical refer- ences to works in English, French, or German, as these are the three languages of which those interested in mathematics are most likely to have an adequate reading knowledge. For those especially interested in geometry, Italian also is necessary. Italian historical works are included in the bibliographical material of the histories listed. To the many professional friends who have advised me on their respective specialties and whose generous help I have attempted to pass on to others, I am very grateful. A special word of thanks is due Professor W. H. Gage, of the University of British Columbia, who removed many obscurities and greatly improved several of the presentations. This has been an opportunity to do something a little off the beaten track to show prospective readers how the mathe- matics familiar to them got where it is, and where it is going TO ANY PROSPECTI TE READER xi from there. I trust that students will tolerate the departure from the traditional textbook. For one thing, at any rate, the more sensible should be grateful: only the most ingenious instructor could set an examination on the book. It has, unhappily, been necessary in writing the book to consider many things besides the masterpieces of mathematics. Rising from a protracted and not always pleasant session with the works of bickering historians, scholarly pedants, and con- tentious mathematicians, often savagely contradicting or meanly disparaging one another, I pass on, for what it may be worth, the principal tiling I liave learned to appreciate as never before. It is contained in Buddha’s last injunction to his followers: Believe nothing on hearsay. Do not believe in traditions because they are old, or in anything on the mere authority of myself or any other teacher. E. T. Bell. Note to the Second Edition About fifty pages of new material have been added in this edition. The additions include numerous short amplifications of miscellaneous topics from Greek mathematics to mathematical logic, with longer notes on symbolism, algebraic and differential geometry, lattices, and other subjects in which there have been striking recent advances. E. T. Bell. California Institute of Technology, Pasadena, California, July, ms. Contents Pack To Any Prospective Reader v Ctumat 1. General Prospectus 3 2. The Age of Empiricism 26 3. Firmly Established 49 4. The European Depression 85 5. Detour through India, Arabia, and Spain 93 6. Four Centuries of Transition, 1202-1603 107 7. The Beginning of Modem Mathematics, 1637-1687 131 8. Extensions of Number 167 9. Toward Mathematical Structure 186 10. Arithmetic Generalized 218 11. Emergence of Structural Analysis 245 12. Cardinal and Ordinal to 1902 270 13. From Intuition to Absolute Rigor, 1700-1900 282 14. Rational Arithmetic after Fermat 296 15. Contributions from Geometry 320 16. The Impulse from Science 361 17. From Mechanics to Generalized Variables 370 IS. From Applications to Abstractions 382 19. Differential and Difference Equations 400 20. Invariance 420 21. Certain Major Theories of Functions 469 22. Through Physics to General Analysis and Abstractness. . . 516 23. Uncertainties and Probabilities 548 Notes. 595 Index 611 xin The Development of Mathematics CHAPTER 1 General Prospectus In all historic times all civilized peoples have striven toward mathematics. The prehistoric origins are as irrecoverable as those of language and art, and even the civilized beginnings can only be conjectured from the behavior of primitive peoples today. Whatever its source, mathematics has come down to the present by the two main streams of number and form. The first carried along arithmetic and algebra, the second, geometry. In the seven- teenth century these two united, forming the ever-broadening river of mathematical analysis. We shall look back in the following chapters on this great river of intellectual progress and, in the diminishing perspective of time, endeavor to see the more outstanding of those elements in the general advance from the past to the present which have endured. ‘Form,’ it may be noted here to prevent a possible misappre- hension at the outset, has long been understood mathematically in a sense more general than that associated with the shapes of plane figures and solid bodies. The older, geometrical meaning is still pertinent. The newer refers to the structure of mathematical relations and theories. It developed, not from a study of spacial form as such, but from an analysis of the proofs occurring in geometry, algebra, and other divisions of mathematics. Awareness of number and spacial form is not an exclusively human privilege. Several of the higher animals exhibit a rudi- mentary sense of number, while -others approach genius in their mastery of form. Thus a certain cat made no objection when she was relieved of two of her six kittens, but was plainly distressed when she was deprived of three. She was relatively as advanced arithmetically as the savages of an Amazon tribe who can count 3 4 THE DEVELOPMENT OF MATHEMATICS up to two, but who confuse all greater numbers in a nebulous ‘many.’ Again, the intellectual rats that find their way through the mazes devised by psychologists are passing difficult examinations in topology. At the human level, a classic puzzle which usually suffices to show the highly intelligent the limitations of their spacial intuition is that of constructing a surface with only one side and one boundary. Although human beings and the other animals thus meet on a common ground of mathematical sense, mathematics as it has been understood for at least twenty-five centuries is on a far higher plane of intelligence. Necessity for proof; emergence of mathematics Between the workable empiricism of the early land measurers who parceled out the fields of ancient Egypt and the geometry of the Greeks in the sixth century before Christ there is a great chasm. On the remoter side lies what preceded mathematics, on the nearer, mathematics; and the chasm is bridged by deductive reasoning applied consciously and deliberately to the practical inductions of daily life. Without the strictest deductive proof from admitted assumptions, explicitly stated as such, mathe- matics does not exist. This does not deny that intuition, experi- ment, induction, and plain guessing are important elements in mathematical invention. It merely states the criterion by which the final product of all the guessing, by whatever name it be dignified, is judged to be or not to be mathematics. Thus, for example, the useful rule, known to the ancient Babylonians, that the area of a rectangular field can be computed by ‘length times breadth,’ may agree with experience to the utmost refinement of physical measurement; but the rule is not a part of mathematics until it has been deduced from explicit assumptions. It may be significant to record that this sharp distinction between mathematics and other sciences began to blur slightly under the sudden impact of a greatly accelerated applied mathe- matics, so called, in the second world war. Semiempirical pro- cedures of calculation, certified by their pragmatic utility in war, were accorded full mathematical prestige. This relaxation of traditional demands brought the resulting techniques closer in both method and spirit to engineering and the physical sciences. It was acclaimed by some of its practitioners as a long-overdue democratization of the most aristocratic of the GENERAL PROSPECTUS 5 sciences. Others, of a more conservative persuasion, deplored the passing of the ideal of strict deduction, as a profitless con- fusion of a simple issue which had at last been clarified after several centuries of futile disputation. One fact, however, emerged from the difference of opinion: It is difficult, in modern warfare, to wreck, to maim, or to kill efficiently without a con- siderable expenditure of mathematics, much of which was designed originally for the development of those sciences and arts which create and conserve rather than destroy and waste. It is not known where or when the distinction between induc- tive inference — the summation of raw experience — and deduc- tive proof from a set of postulates was first made, but it was sharply recognized by the Greek mathematicians as early as 550 b.c. As will appear later, there may be some grounds for believing that the Egyptians and the Babylonians of about 2000 B.c. had recognized the necessity for deductive proof. For proof in even the rough and unready calculations of daily life is indeed a necessity, as may be seen from the mensuration of rectangles. If a rectangle is 2 feet broad and 3 long, an easy proof sus- tains the verdict of experience, founded on direct measurement, that the area is 6 square feet. But if the breadth is ‘\j2 and the length "\/3 feet, the area cannot be determined as before by cutting the rectangle into unit squares; and it is a profoundly difficult problem to prove that the area is V6 feet, or even to give intelligible, usable meanings to \^2, VI j '\J6, and ‘area.’ By taking smaller and smaller squares as unit areas, closer and closer approximations to the area are obtained, but a barrier is soon reached beyond which direct measurement cannot proceed. This raises a question of cardinal importance for a just under- standing of the development of all mathematics, both pure and applied. Continuing with the \^2 X VI rectangle, we shall suppose that refined measurement has given 2.4494897 as the area. This is correct to the seventh decimal, but it is not right, because ‘\J6, the exact area, is not expressible as a terminated decimal frac- tion. If seven-place accuracy is the utmost demanded, the area has been found. This degree of precision suffices for many practical applications, including precise surveying. But it is inadequate for others, such as some in the physical sciences and modern statistics. And before the seven-place approximation 6 THE DEVELOPMENT OF MATHEMATICS can be used intelligently, its order of error must be ascertained. Direct measurement cannot enlighten us; for after a certain limit, quickly passed, all measurements blur in a common uncer- tainty. Some universal agreement on what is meant by the exact area must be reached before progress is possible. Experience, both practical and theoretical, has shown that a consistent and useful mensuration of rectangles is obtained when the rule ‘length times breadth’ is deduced from postulates abstracted from a lower level of experience and accepted as valid. The last is the methodology of all mathematics. Mathematicians insist on deductive proof for practically workable rules obtained inductively because they know that analogies between phenomena at different levels of experience are not to be accepted at their face value. Deductive reasoning is the only means yet devised for isolating and examining hidden assumptions, and for following the subtle implications of hypoth- eses which may be less factual than they seem. In its modern technical uses of the deductive method, mathematics employs much sharper tools than those of the traditional logic inherited from ancient and medieval times. Proof is insisted upon for another eminently practical reason. The difficult technology of today is likely to become the easy routine of tomorrow; and a vague guess about the order of magnitude of an unavoidable error in measurement is worthless in the technological precision demanded by modern civilization. Working technologists cannot be skilled mathematicians. But unless the rules these men apply in their technologies have been certified mathematically and scientifically by competent experts, they are too dangerous for use. There is still another important social reason for insistence on mathematical demonstration, as may be seen again from the early history of surveying. In ancient Egypt, the primitive theory of land measurement, without which the practice would have been more crudely wasteful than it actually was, sufficed for the economy of the time. Crude both practically and theoreti- cally thor ~ti this surveying was, it taxed the intelligence of the Egyptian, mathematicians. Today the routine of precise survey- ing can be mastered by a boy of seventeen; and those applica- tions of the trigonometry that evolved from primitive surveying and astronomy which are of greatest significance in our own civilization have no connection with surveying. Some concern mechanics and electrical technology, others, the most advanced GENERAL PROSPECTUS 7 parts of the physical sciences from which the industries of twenty or a hundred years hence may evolve.^, Now, contrary to what might be supposed, modern trigonom- etry did not develop in response to any practical need. Modern trigonometry is impossible without the calculus and the mathe- matics of V~l- To cite but one of the commoner applications, over a century and a half elapsed before this trigonometry became indispensable in the theory and practice of alternating currents. Long before anyone had dreamed of an electric dynamo, the necessary mathematics of dynamo design was available. It had developed largely because the analysts of the eighteenth century sought to understand mathematically the somewhat meager legacy of trigonometry bequeathed them by the astron- omers of ancient Greece, the Hindus, and the mathematicians of Islam. Neither astronomy nor any other science of th e eight- eenth century suggested the introduction of V~l> which completed trigonometry, as no such science ever made any use of the finished product. The importance of mathematics, from Babylon and Egypt to the present, as the primary source of workable approximations to the complexites of daily life is generally appreciated. In fact, a mathematician might believe it is almost too generally appre- ciated. It has been preached at the public, in school and out, by socially conscious educators until almost anyone may be par- doned for believing that the rule of life is rule of thumb. Because routine surveying, say, requires only mediocre intelligence, and because surveying is a minor department of applied mathe- matics, therefore only that mathematics which can be manipu- lated by rather ordinary people is of any social value. But no growing economy can be sustained by rule of thumb. If new applications of a furiously expanding science are to be possible, difficult and abstruse mathematical theories far beyond the college level must continue to be dcveloped,by those having the requisite talents. In this living mathematics it is imagination and rigorous proof which count, not the numerical accuracy of the machine shop or the computing laboratory. A familiar example from common things will show the neces- sity for mathematics as distinguished from calculation. A nautical almanac is one of the indispensablcs of modern naviga- tion and hence of commerce. Machines are now commonly used for the heavy labor of computing. Ultimately the computations depend upon the motions of the planets, and these are calculated 8 THE DEVELOPMENT OF MATHEMATICS from the infinite (non-terminating) series of numbers given by the Newtonian theory of gravitation. For the actual work of computation a machine is superior to any human brain; but no machine yet invented has had brains enough to reject nonsense fed into it. From a grotesquely absurd set of data the best of machines will return a final computation that looks as reasonable as any other. Unless the series used in dynamical astronomy converge to definite limiting numbers (asymptotic series also are used, but not properly divergent), it is futile to calculate by means of them. A table computed by properly divergent series would be indistinguishable to the untrained eye from any other; but the aviator trusting it for a flight from Boston to New York might arrive at the North Pole. Despite its inerrant accuracy and attractive appearance, even the most highly polished mechanism is no substitute for brains. The research mathe- matician and the scientific engineer supply the brains; the machine does the rest. Nobody with a grain of common sense would demand a strict proof for every tentative application of complicated mathe- matics to new situations. Occasionally in problems of excessive difficulty, like some of those in nuclear physics, calculations are performed blindly without reference to mathematical validity; but even the boldest calculator trusts that his temerity will some day be certified rationally. This is a task for the mathe- maticians, not for the scientists. And if science is to be more than a midden of uncorrelated facts, the task must be carried through. Necessity for abstractness With the recognition that strict deductive reasoning has both practical and aesthetic values, mathematics began to emerge some six centuries before the Christian era. The emergence was complete when human beings realized that common experience is too complex for accurate description. Again it is not known when or where this conclusion was first reached, but the Greek geometers of the fourth century b.c. at latest had accepted it, as is shown by their work. Thus Euclid in that century stated the familiar definition: “ A circle is a plane figure contained by one line, called the circumference, and is such that all straight lines drawn from a certain point, called the center, within the figure to the circumference are equal.” There is no record of any such figure as Euclid’s circle ever having been observed by any human being. Yet Euclid’s ideal GENERAL PROSPECTUS 9 circle is not only that of school geometry, but is also the circle of the handbooks used by engineers in calculating the performance of machines. Euclid’s mathematical circle is the outcome of a deliberate simplification and abstraction of observed disks, like the full moon’s, which appear ‘circular’ to unaided vision. This abstracting of common experience is one of the principal sources of the utility of mathematics and the secret of its scien- tific power. The world that impinges on the senses of all but introverted solipsists is too intricate for any exact description yet imagined by human beings. By abstracting and simplifying the evidence of the senses, mathematics brings the worlds of science and daily life into focus with our myopic comprehension, and makes possible a rational description of our experiences which accords remarkably "well with observation. Abstractness, sometimes hurled as a reproach at mathe- matics, is its chief glory and its surest title to practical useful- ness. It is also the source of such beauty as may spring from mathematics. History and proof In any account of the development of mathematics there is a peculiar difficulty, exemplified in the two following assertions, about many statements concerning proof. ( A ) It is proved in Proposition 47, Book 1, of Euclid’s Ele- ments, that the square on the longest side of a right-angled tri- angle is equal to the sum of the squares on the other two sides (the so-called Pythagorean theorem). ( B ) Euclid proved the Pythagorean theorem in Proposition 47 of Book I of his Elements. In ordinary discourse, ( A ), ( B ) would usually be considered equivalent — both true or both false. Here ( A ) is false and ( B ) true. For a clear understanding of the development of mathe- matics it is important to see that this distinction is not a quibble. It is also essential to recognize that comprehension here is more important than knowing the date (c. 330-320 b.c.) at which the Elements were written, or any other detail of equal antiquarian interest. In short, the crux of the matter is mathematics, which is at least as important as history, even in histories of mathematics. The statement ( A) is false because the attempted proof in the Elements is invalid. The attempt is vitiated by tacit assumptions that Euclid ignored in laying down the postulates from which he undertook to deduce the theorems in his geometry. From those 10 THE DEVELOPMENT OF MATHEMATICS same postulates it is easy to deduce, by irrefragable logic, spec- tacularly paradoxical consequences, such as “all triangles are equilateral.” Thus when an eminent scholar of Greek mathe- matics asserts that owing to the “inerring logic” of the Greeks, “there has been no need to reconstruct, still less to reject as unsound, any essential part of their doctrine,” mathematicians must qualify assent by referring to the evidence. The “essential part of their doctrine” has indeed come down to us unchanged, that part being insistence on deductive proof. But in the specific instance of Euclid’s proofs, many have been demolished in detail, and it would be easy to destroy more were it worth the trouble. The statement (B) is true because the validity of a proof is a function of time. The standard of mathematical proof has risen steadily since 1821, and finality is no longer sought or desired. In Euclid’s day, and for centuries thereafter, the attempted proof of the Pythagorean proposition satisfied all the current requirements of logical and mathematical rigor. A sound proof today does not differ greatly in outward appearance from Euclid’s; but if we inspect the postulates required to validate the proof, we notice several which Euclid overlooked. A carefully taught child of fourteen today can easily detect fatal omissions in many of the demonstrations in elementary geometry accepted as sound less than fifty years ago. It is clear that we must have some convention regarding ‘proof.’ Otherwise, few historical statements about mathematics will have any meaning. Whenever in the sequel it is stated that a certain result was proved, this is to be understood for the sense as in ( B ), namely, that the proof was accepted as valid by professional mathematicians at the time it was given. If, for example, it is asserted that a work of Newton or of Euler contains a proof of the binomial theorem for exponents other than positive integers, the assertion is false for the ( A ) meaning, true for the (. B ). The proofs which these great mathematicians gave in the seventeenth and eighteenth centuries were valid at that time, although they would not be accepted today by a competent teacher from a student in the first college course. It need scarcely be remarked that few modest mathemati- cians today expect all of their own proofs to survive the criticisms of their successors unscathed. Mathematics thrives on intelli- gent criticism, and it is no disparagement of the great work of the past to point out that its very defects have inspired work as great. GENERAL PROSPECTUS 11 Failure to observe that mathematical validity depends upon its epoch may generate scholarly but vacuous disputes over historical minutiae. Thus a meticulous historian who asserts that the Greeks of Euclid’s time failed to solve quadratic equations by their geometric method because they ‘overlooked’ possible negative roots, to say nothing of imaginaries, himself overlooks one of the most interesting phenomena in the entire history of mathematics. Until positive rational fractions and negative numbers were invented by mathematicians (or ‘discovered,’ if the inventors happened to be Platonic realists), a quadratic equation with rational integer coefficients had precisely one root, or precisely two, or precisely none. A Babylonian of a sufficiently remote century who gave 4 as the root of x- — x + 12 had solved his equation completely, because — 3, which we now say is the other root, did not exist for him. Negative numbers were not in his number system. The successive enlargements of the number system necessary to provide all algebraic equations with roots equal in number to the respective degrees of the equations was one of the outstanding landmarks in mathematical progress, and it took about four thousand years of civilized mathematics to establish it. The final necessary extension was delayed till the nineteenth century. An educated algebraist today, wishing to surpass the meticu- lous critic in pedantry, would point out that “how many roots has x 2 = x ?” is a meaningless question until the domain in which the roots may lie has been specified. If the domain is that of complex numbers, this equation has precisely two roots, 0, 1. But if the domain is that of Boolean algebra, this same quadratic (since 1847) has had n roots where « is any integer equal to, or greater than, 2. Boolean algebra, it may be remarked, is as legiti- mately a province of algebra today as is the theory of quadratic equations in elementary schoolbooks. In short, criticizing our predecessors because they completely solved their problems within the limitations which they themselves imposed is as pointless as deploring our own inability to imagine the mathe- matics of seven thousand years hence. Some of the most significant episodes in the entire history of mathematics will be missed unless this dependence of validity upon time is kept in mind as we proceed. In ancient Greece, for example, the entire development of by far the greater part of such Greek mathematics as is still of vital interest stems from 12 THE DEVELOPMENT OF MATHEMATICS this fact. The discontinuities in the time curve of acceptable proof, where standards of rigor changed abruptly, are perhaps the points of greatest interest in the development of mathe- matics. The four most abrupt appear to have been in Greece in the fifth century b.c., in Europe in the 1820’s and in the 1870’s, and again in Europe in the twentieth century. None of this implies that mathematics is a shifting quicksand. Mathematics is as stable and as firmly grounded as anything in human experience, and far more so than most things. Euclid’s Proposition I, 47 stands, as it has stood for over 2,200 years. Under the proper assumptions it has been rigorously proved. Our successors may detect flaws in our reasoning and create new mathematics in their efforts to construct a proof satisfying to themselves. But unless the whole process of mathematical devel- opment suffers a violent mutation, there will remain some proposition recognizably like that which Euclid proved in his generation. Not all of the mathematics of the past has survived, even in suitably modernized form. Much has been discarded as trivial, inadequate, or cumbersome, and some has been buried as defi- nitely fallacious. There could be no falser picture of mathematics than that of “the science which has never had to retrace a step.” If that were true, mathematics would be the one perfect achieve- ment of a race admittedly incapable of perfection. Instead of this absurdity, we shall endeavor to portray mathematics as the constantly growing, human thing that it is, advancing in spite of its errors and partly because of them. Five streams The picture will be clearer if its main outlines are first roughly blocked in and retained while details are being inspected. Into the two main streams of number and form flowed many tributaries. At first mere trickles, some quickly swelled to the dignity of independent rivers. Two in particular influenced the whole course of mathematics from almost the earliest recorded history to the twentieth century. Counting by the natural num- bers 1, 2, 3, . . . introduced mathematicians to the concept of discreteness. The invention of irrational numbers, such as A/2, V3, V6; attempts to compute plane areas bounded by curves or by incommensurable straight lines; the like for surfaces and volumes; also a long struggle to give a coherent account of GENERAL PROSPECTUS 13 motion, growth, and other sensually continuous change, forced mathematicians to invent the concept of continuity. The whole of mathematical history may be interpreted as a battle for supremacy between these two concepts. This conflict may be but an echo of the older strife so prominent in early Greek philosophy, the struggle of the One to subdue the Many. But the image of a battle is not wholly appropriate, in mathematics at least, as the continuous and the discrete have frequently helped one another to progress. One type of mathematical mind prefers the problems asso- ciated with continuity. Geometers, analysts, and appliers of mathematics to science and technology are of this type. The complementary type, preferring discreteness, takes naturally to the theory of numbers in all its ramifications, to algebra, and to mathematical logic. No sharp line divides the two, and the master mathematicians have worked with equal ease in both the continuous and the discrete. In addition to number, form, discreteness, and continuity, a fifth stream has been of capital importance in mathematical history, especially since the seventeenth century. As the sciences, beginning with astronomy and engineering in ancient times and ending with biology, psychology, and sociology in our own, be- came more and more exact, they made constantly increasing demands on mathematical inventiveness, and were mainly responsible for a large part of the enormous expansion of all mathematics since 1637. Again, as industry and invention be- came increasingly scientific after the industrial revolution of the late eighteenth and early nineteenth centuries, they too stimu- lated mathematical creation, often posing problems beyond the existing resources of mathematics. A current instance is the problem of turbulent flow, of the first importance in aerody- namics. Here, as in many similar situations, attempts to solve an essentially new technological problem have led to further ex- pansions of pure mathematics. \yS'The timescale It will be well to have some idea of the distribution of mathe- matics in time before looking at^Ynjividual advances. The time curve of mathenhatical productivity is roughly similar to the exponential cum - of biologic growth, starting to rise almost imperceptibly in the Wmote past and shooting up 14 THE DEVELOPMENT OF MATHEMATICS with ever greater rapidity as the present is approached- The curve is by no means smooth; for, like art, mathematics has had its depressions. There was a deep one in the Middle Ages, owing to the mathematical barbarism of Europe being only partly balanced by the Moslem civilization, itself (mathematically) a sharp recession from the great epoch (third century b.c.) of Archimedes. But in spite of depressions, the general trend from the past to the present has been in the upward direction of a steady increase of valid mathematics. ■f We should not expect the curve for mathematics to follow those of other civilized activities, say art and music, too closely. Masterpieces of sculpture once shattered are difficult to restore or even to remember. The greater ideas of mathematics survive and are carried along in the continual flow, permanent additions immune to the accidents of fashion. Being expressed in the one universally intelligible language as yet devised by human beings, the creations of mathematics are independent of national taste, as those of literature are not. Who today except a few scholars is interested or amused by the ancient Egyptian novelette of the two thieves? And how many can understand hieroglyphics suf- ficiently to elicit from the story whatever significance it may once have had for a people dead all of three thousand years ? But tell any engineer, or any schoolboy who has had some mensura- tion, the Egyptian rule for the volume of a truncated square pyramid, and he will recognize it instantly. Not only are the valid creations of mathematics preserved; their mere presence in the stream of progress induces new currents of mathematical thought. jThe majority of working mathematicians acquainted in some measure with the mathematics created since 1800 agree that the time curve rises more sharply thereafter than before. An open mind on this question is necessary for anyone wishing to see mathematical history as the majority of mathematicians see it. Many who have no firsthand knowledge of living mathematics beyond the calculus believe on grossly inadequate evidence that mathematics experienced its golden age in some more or less re- mote past. Mathematicians, think not. The recent era, beginning in the nineteenth century, is usually regarded as the golden age by those personally conversant with mathematics and at least some of its history. An unorthodox but reasonable apportionment of the time- scale of mathematical development cuts all history into three GENERAL PROSPECTUS 15 periods of unequal lengths. These may be called the remote, the middle, and the recent. The remote extends from the earliest times of which we have reliable knowledge to a.d. 1637, the mid- dle from 1638 to 1800. The recent period, that of modern mathe- matics as professionals today understand mathematics, extends from 1801 to the present. Some might prefer 1821 instead of 1801. There are definite reasons for the precise dates. Geometry became analytic in 1637 with the publication of Descartes’ mas- terpiece. About half a century later the calculus of Newton and Leibniz, also the dynamics of Galileo and Newton, began to become the common property of all creative mathematicians. Le'bniz certainly was competent to estimate the magnitude of this advance. He is reported to have said that, of all mathematics from the beginning of the world to the time of Newton, •what Newton had done was much the better half. The eighteenth century exploited the methods of Descartes, Newton, and Leibniz in all departments of mathematics as they then existed. Perhaps the most significant feature of this century was the beginning of the abstract, completely general attack. Although adequate realization of the power of the abstract method was delayed till the twentieth century, there are notable anticipations in Lagrange’s work on algebraic equations and, above all, in his analytic mechanics. In the latter, a direct, uni- versal method unified mechanics as it then was, and has re- mained to this day one of the most powerful tools in the physical sciences. There was nothing like this before Lagrange. The last date, 1801, marks the beginning of a new era of unprecedented inventiveness, opening with the publication of Gauss’ masterpiece. The alternative, 1821, is the year in which Cauchy began the first satisfactory treatment of the differential and integral calculus. As one instance of the greatly accelerated productivity in the nineteenth century, consequent to a thorough mastery and amplification of the methods devised in the middle period, an episode in the development of geometry is typical. Each of five men — Lobachewsky, Bolyai, Plucker, Riemann, Lie — invented as part of his lifework as much (or more) new geometry as was created by all the Greek mathematicians in the two or three centuries of their greatest activity. There are good grounds for the frequent assertion that the nineteenth century alone contrib- uted about five times as much to mathematics as had all pre- 16 THE DEVELOPMENT OF MATHEMATICS ceding history. This applies not only to quantity but, what is of incomparably greater importance, to power. Granting that the mathematicians before the middle period may have encountered the difficulties attendant on all pioneer- ing, we need not magnify their great achievements to universe- filling proportions. It must be remembered that the advances of the recent period have swept up and included nearly all the valid mathematics that preceded 1800 as very special instances of general theories and methods. Of course nobody who works in mathematics believes that our age has reached the end, as La- grange thought his had just before the great outburst of the recent period. But this does not alter the fact that most of our predecessors did reach very definite ends, as we too no doubt shall. Their limited methods precluded further significant prog- ress, and it is possible, let us hope probable, that a century hence our own more powerful methods will have given place to others yet more powerful. Seven periods A more conventional division of the time-scale separates all mathematical history into seven periods : (1) From the earliest times to ancient Babylonia and 'Egypt, inclusive. (2) The Greek contribution, about 600 b.c. to about a.d. 300, the best being in the fourth and third centuries b.c. (3) The oriental and Semitic peoples — Hindus, Chinese, Per- sians, Moslems, Jews, etc., partly before, partly after (2), and extending to (4). (4) Europe during the Renaissance and the Reformation, roughly the fifteenth and sixteenth centuries. (5) The seventeenth and eighteenth centuries. (6) The nineteenth century. (7) The twentieth century. This division follows loosely the general development of Western civilization and its indebtedness to the Near East. Possibly (6), (7) are only one, although profoundly significant new trends became evident shortly after 1900. In the sequel, we Ghall observe what appears to have been the main contribution in each of the seven periods. A few anticipatory remarks here may clarify the picture for those seeing it for the first time. Although the peoples of the Near East were more active than the Europeans during the third of the seven periods, mathe- GENERAL PROSPECTUS 17 matics as it exists today is predominantly a product of Western civilization. Ancient advances in China, for example, either did not enter the general stream or did so by commerce not yet traced. Even such definite techniques as were devised either belong to the trivia of mathematics or were withheld from European mathematicians until long after their demonstrably independent invention in Europe. For example, Horner’s method for the numerical solution of equations may have been known to the Chinese, but Horner did not know that it was. And, as a matter of fact, mathematics would not be much the poorer if neither the Chinese nor Horner had ever hit on the method. European mathematics followed a course approximately parallel to that of the general culture in the several countries. Thus the narrowly practical civilization of ancient Rome con- tributed nothing to mathematics; when Italy was great in art, it excelled in algebra; when the last surge of the Elizabethan age in England had spent itself, supremacy in mathematics passed to Switzerland and France. Frequently, however, there were sporadic outbursts of isolated genius in politically minor coun- tries, as in the independent creation of non-Euclidean geometry in Hungary in the early nineteenth century. Sudden upsurges of national vitality were occasionally accompanied by increased mathematical activity, as in the Napoleonic wars following the French Revolution, also in Germany after the disturbances of 1848. But the world war of 1914—18 appears to have been a brake on mathematical progress in Europe and to a lesser degree elsewhere, as also were the subsequent manifestations of na- tionalism in Russia, Germany, and Italy. These events hastened the rapid progress which mathematics had been making since about 1890 in the United States of America, thrusting that coun- try into a leading position. The correlation between mathematical excellence and bril- liance in other aspects of general culture was sometimes negative. Several instances might be given; the most important for the development of mathematics falls in the Middle Ages. When Gothic architecture and Christian civilization were at their zenith in the twelfth century (some would say in the thirteenth), European mathematics was just beginning the ascent from its nadir. It will be extremely interesting to historians eight cen- turies hence if it shall appear that the official disrepute into which mathematics and impartial science had fallen in certain European countries some years before the triumph of medieval 18 THE DEVELOPMENT OF MATHEMATICS ideals in September, 1939, was the dawn of a new faith about to enshrine itself in the unmathematical simplicities of a science- less architecture. Our shaggy ancestors got along for hundreds of thousands of years without science or mathematics in their filthy caves, and there is no obvious reason why our brutalized descendants — if they are to be such — should not do the same. Attending here only to acquisitions of the very first magni- tude in all seven of the periods, we may signalize three. All will be Rioted in some detail later. KJ The most enduringly influential contribution to mathematics of all the periods prior to the Renaissance was the Greek inven- tion of strict deductive reasoning. Next in mathematical impor- tance is the Italian and French development of symbolic algebra during the Renaissance. The Hindus of the seventh to the twelfth centuries a.d. had almost invented algebraic symbolism; the Moslems reverted in their classic age to an almost completely rhetorical algebra. The third major advance has already been indicated, but may be emphasized here: in the earlier part of the fifth period — seventeenth century — the three main streams of number, form, and continuity united. This generated the calculus and mathematical analysis in general; it also trans- formed geometry and made possible the later creation of the higher spaces necessary for modern applied mathematics. The leaders here were French, English, and German. • The fifth period is usually considered as the fountainhead of modern pure mathematics. It brackets the beginning of modern science; and another major advance was the extensive applica- tion of the newly created pure mathematics to dynamical astronomy, following the work of Newton, and, a little later, to the physical sciences, following the methodology of Galileo and Newton. Finally, in the nineteenth century, the great river burst its banks, deluging wildernesses where no mathematics had flourished and making them fruitful. . If the mathematics of the twentieth century differs signifi- cantly from that of nineteenth, possibly the most important dis- tinctions are a marked increase in abstractness with a consequent gain in generality, and a growing preoccupation with the mor- phology and comparative anatomy of mathematical structures; a sharpening of critical insight; and a dawning recognition of the limitations of classical deductive reasoning. If ‘limitations’ sug- gests frustration after about seven thousand years of human strivings to think clearly, the suggestion is misleading. But it is GENERAL PROSPECTUS 19 true that the critical evaluations of accepted mathematical reasoning which distinguished the first four decades of the twentieth century necessitated extensive revisions of earlier mathematics, and inspired much new work of profound interest for both mathematics and epistemology. They also led to what appeared to be the final abandonment of the theory that mathe- matics is an image of the Eternal Truth. . The division of mathematical history into about seven periods is more or less traditional and undoubtedly is illuminat- ing, especially in relation to the fluctuating light which we call civilization. But the unorthodox remote, middle, and recent periods, described earlier, seem to give a truer presentation of the development of mathematics itself and a more vivid sugges- tion of its innate vitality. • Some general characteristics In each of the seven periods there was a well-defined rise to maturity and a subsequent decline in each of several limited modes of mathematical thought. Without fertilization by crea- tive new ideas, each was doomed to sterility. In the Greek period, for example, synthetic metric geometry, as a method, got as far as seems humanly possible with our present mental equipment. It was revivified into something new by the ideas of analytic geometry in the seventeenth century, by those of projective geometry in the seventeenth and nineteenth centuries, and finally, in the eighteenth and nineteenth centuries, by those of differential geometry. Such revitalizations were necessary not only for the con- tinued growth of mathematics but also for the development of science. Thus it would be impossible for mathematicians to apprehend the subtle complexities of the geometries applied to modern science by the methods of Euclid and Apollonius. And in pure mathematics, much of the geometry of the nineteenth century was thrust aside by the more vigorous geometries of abstract spaces and the non-Riemannian geometries developed in the twentieth. Considerably less than forty years after the close of the nineteenth century, some of the geometrical master- pieces of that heroic age of geometry were already beginning to seem otiose and antiquated. This appears to be the case for much of classical differential geometry and synthetic projective geome- try. If mathematics continues to advance, the new geometries 20 THE DEVELOPMENT OF MATHEMATICS of the twentieth century will likely be displaced in their turn, or be subsumed under still rarer abstractions. In mathematics, of all places, finality is a chimera. Its rare appearances are witnessed only by the mathematically dead. As a period closes, there is a tendency to overelaboration of merely difficult things which the succeeding period either ignores as unlikely to be of lasting value, or includes as exercises in more powerful methods. Thus a host of special curves investigated with astonishing vigor and enthusiasm by the early masters of analytic geometry live, if at all, only as problems in elementary textbooks. Perhaps the most extensive of all mathematical cemeteries are the treatises which perpetuate artificially difficult problems in mechanics to be worked as if Lagrange, Hamilton, and Jacobi had never lived. Again, as we approach the present, new provinces of mathe- matics are more and more rapidly stripped of their superficial riches, leaving only a hypothetical mother lode to be sought by the better-equipped prospectors of a later generation. The law of diminishing returns operates here in mathematics as in economics: without the introduction of radically new improve- ments in method, the income does not balance the outgo. A conspicuous example is the highly developed theory of algebraic invariants, one of the major acquisitions of the nineteenth cen- tury; another, the classical theory of multiply periodic functions, of the same century. The first of these contributed indirectly to the emergence of general relativity; the second inspired much work in analysis and algebraic geometry. A last phenomenon of the entire development may be noted. At first the mathematical disciplines were not sharply defined. As knowledge increased, individual subjects split off from the parent mass and became autonomous. Later, some were over- taken and reabsorbed in vaster generalizations of the mass from which they had sprung. Thus trigonometry issued from survey- ing, astronomy, and geometry only to be absorbed, centuries later, in the analysis which had generalized geometry. This recurrent escape and recapture has inspired some to dream of a final, unified mathematics which shall embrace all. Early in the twentieth century it was believed by some for a time that the desired unification had been achieved in mathe- matical logic. But mathematics, too irrepressibly creative to be restrained by any formalism, escaped. GENERAL PROSPECTUS 21 Motivation in mathematics Several items in the foregoing prospectus suggest that much of the impulse behind mathematics has been economic. In the third and fourth decades of the twentieth century, for obvious political reasons, attempts were made to show that all vital mathematics, particularly in applications, is of economic origin. To overemphasize the immediately practical in the develop- ment of mathematics at the expense of sheer intellectual curiosity is to miss at least half the fact. As any moderately competent mathematician whose education has not stopped short with the calculus and its commoner applications may verify for himself, it simply is not true that the economic motive has been more frequent than the purely intellectual in the creation of mathematics. This holds for practical mathematics as applied in commerce, including all insurance, science, and the technologies, as well as for those divisions of mathematics which at present are economically valueless. Instances might be multiplied indefi- nitely; four must suffice here, one from the theory of numbers, two from geometry, and one from algebra. About twenty centuries before the polygonal numbers were generalized, and considerably later applied to insurance and to statistics, in both instances through combinatorial analysis, the former by way of the mathematical theory of probability, their amusing peculiarities were extensively investigated by arithme- ticians without the least suspicion that far in the future these numbers were to prove useful in practical affairs. The polygonal numbers appealed to the Pythagoreans of the sixth century b.c. and to their bemused successors on account of the supposedly mystical virtues of such numbers. The impulse here might be called religious. Anyone familiar with the readily available history of these numbers and acquainted with Plato’s dialogues can trace for himself the thread of number mysticism from the crude numerology of the Pythagoreans to the Platonic doctrine of Ideas. None of this greatly resembles insurance or statistics, v.' Later mathematicians, including one of the greatest, regarded these numbers as legitimate objects of intellectual curiosity. Fermat, cofoundcr with Pascal in the seventeenth century of the mathematical theory of probability, and therefore one of the grandfathers of insurance, amused himself with the polygonal 12 THE DEVELOPMENT OF MATHEMATICS and figurate numbers for years before either he or Pascal ever dreamed of defining probability mathematically. As a second and somewhat hackneyed instance, the conic sections were substantially exhausted by the Greeks about seven- teen centuries before their applications to ballistics and astron- omy, and through the latter to navigation, were suspected. These applications might have been made without the Greek geometry, had Descartes’ analytics and Newton’s dynamics been available. But the fact is that by heavy borrowings from Greek conics the right way was first found. Again the initial motive was intellec- tual curiosity. The third instance is that of polydimensional space. In analytic geometry, a plane curve is represented by an equation containing two variables, a surface by an equation containing three. Cayley in 1843 transferred the language of geometry to systems of equations in more than three variables, thus inventing a geometry of any finite number of dimensions. This generaliza- tion was suggested directly by the formal algebra of common analytic geometry, and was elaborated for its intrinsic interest before uses for it were found in thermodynamics, statistical mechanics, and other departments of science, including statistics, both theoretical and industrial, as in applied physical chemistry. In passing, it may be noted that one method in statistical me- chanics makes incidental use of the arithmetical theory of partitions, which treats of such problems as determining in how many ways a given positive integer is a sum of positive integers. This theory was initiated by Euler in the eighteenth century, and for over 150 years was nothing but a plaything for experts in the perfectly useless theory of numbers. The fourth instance concerns abstract algebra as it has developed since 1910. Any modern algebraist may easily verify that much of his work has a main root in one of the most fantas- tically useless problems ever imagined by curious man, namely, in Fermat’s famous assertion of the seventeenth century that x n + y n — zP is impossible in integers x, y, z all different from zero if n is an integer greater than two. Some of this recent algebra quickly found use in the physical sciences, particularly in modern quantum mechanics. It was developed without any suspicion that it might be scientifically useful. Indeed, not one of the algebraists concerned was competent to make any signifi- cant application of his work to science, much less to foresee that such applications would some day be possible. As late as the GENERAL PROSPECTUS 23 autumn of 1925, only two or three physicists in the entire world had any inkling of the new channel much of physics was to follow in 1926 and the succeeding decade. Residues of epochs In following the development of mathematics, or of any science, it is essential to remember that although some partic- ular work may now be buried it is not necessarily dead. Each epoch has left a mass of detailed results, most of which are now of only antiquarian interest. For the remoter periods, these survive as curiosities in specialized histories of mathematics. For the middle and recent periods — since the early decades of the seventeenth century — innumerable theorems and even highly developed theories are entombed in the technical journals and transactions of learned societies, and are seldom if ever men- tioned even by professionals. The mere existence of many is all but forgotten. The lives of thousands of workers have gone into this moribund literature. In what sense do these half-forgotten things live r And how can it be truthfully said that the labor of all those toilers was not wasted ? The answers to these somewhat discouraging questions are obvious to anyone who works in mathematics. Out of all the uncoordinated details at last emerges a general method or a new concept. The method or the concept is what survives. By means of the general method the laborious details from which it evolved arc obtained uniformly and with comparative ease. The new concept is seen to be more significant for the whole of mathe- matics than are the obscure phenomena from which it was abstracted. But such is the nature of the human mind that it almost invariably takes the longest way round, shunning the straight road to its goal. There is no principle of least action in scientific discovery. Indeed, the goal in mathematics frequently is unperccivcd until some explorer more fortunate than his rivals blunders onto it in spite of his human inclination to follow the crookcdest path. Simplicity and directness are usually the last things to be attained. In illustration of these facts we may cite once more the theory of algebraic invariants. When this theory was first de- veloped in the nineteenth century, scores of devoted workers slaved at the detailed calculation of particular invariants and covariants. Their work is buried. But its very complexity drove their successors in algebra to simplicity: masses of apparently 24 THE DEVELOPMENT OF MATHEMATICS isolated phenomena were recognized as instances of simple underlying general principles. Whether these principles would ever have been sought, much less discovered, without the urge imparted by the massed calculations, is at least debatable. The historical fact is that they were so sought and discovered. In saying that the formidable lists of covariants and invar- iants of the early period are buried, we do not mean to imply that they are permanently useless; for the future of mathematics is as unpredictable as is that of any other social activity. But the methods and principles of the later period make it possible to obtain all such results with much greater ease should they ever be required, and it is a waste of time and effort today to add to them. One residue of all this vast effort is the concept of invariance. So far as can be seen at present, invariance is likely to be il- luminating in both pure and applied mathematics for many decades to come. In our survey we shall endeavor to observe the methods and the concepts which have been sublimated from other masses of details, and which offer similar prospects of endurance. It is not epochs that matter, but their residues. Nor, as epochs recede into the past, do the men who made them obscure the permanence and impersonality of their work with their hopes, their fears, their jealousies, and their petty quarrels. Some of the greatest things that were ever done in mathematics are wholly anonymous. We shall never know who first imagined the numbers 1, 2, 3, . . . , or who first perceived that a single ‘three’ isolates what is common to three goads, three oxen, three gods, three altars, and three men. Two recent opinions on the general history of science are apposite for that of mathematics, and may stand here as an introduction to what is to follow. In his Autobiograpliia (1923), the Spanish histologist Santiago Ramon y Cajal had this to say of scientific history: In spite of all the allegations of self-love, the facts at first associated with the name of a particular man end by being anonymous, lost forever in the ocean of Universal Science, Thus thij monograph imbued with individual human quality becomes incorporated, stripped of sentimentalisms, in the ab- stract doctrine of the general treatise. To the hot sun of actuality will succeed — if they do succeed — the cold beams /si the history of learning. The next is singularly pdrtinent, coming as it does from the man who advanced beyond Newton in the mathematical theory GENERAL PROSPECTUS 25 of gravitation. Speaking of Newton’s w r ork in optics, Einstein says : Newton’s age has long since passed through the sieve of oblivion, the doubtful striving and suffering of his generation have vanished from our ken; the works of some few great thinkers and artists have remained, to delight and ennoble those who come after us. Newton’s discoveries have passed into the stock of accepted knowledge. Finally, we shall try to observe the caution suggested in the observation of an M.D. and writer who is not a mathematician, Halladay Sutherland: “There is always the danger of seeing the past in the light of a golden sunset.” CHAPTER 2 The Age of Empiricism It is not known where, when, or by whom, it was first per- ceived that a mastery of number and form is as useful as lan- guage for civilized living. The historical record begins, in Egypt and in Mesopotamia (Babylonia, including Sumer and Akkad), with both number and form far advanced beyond the primitive stage of culture, and even here the cardinal dates have been disputed. Those dates are 4241 ± 200 b.c. at the earliest and 2781 b.c. at the latest for Egypt, 1 and about 5700 b.c. for Mesopotamia. Both refer to the earliest calendric reckoning, and each is more or less substantiated by astronomical evidence. The basis of both the Egyptian and the Mesopotamian civi- lizations was agriculture. In an agricultural economy a reliable calendar is a necessity. A calendar implies both astronomical and arithmetical accuracy far beyond the facilities of mythology and haphazard observation, and it is not come at in a year. Some primitive peoples who have never been driven to farming have only the vaguest notions of the connection between the periodi- city of seasons and the aspect of the heavens. By 5700 b.c. the Sumerian predecessors of the Semitic Babylonians were dating the beginning of their year from the vernal equinox. A thousand years later the first month of the year was named after the Bull, the sun being in the constellation Taurus at the vernal equinox of about 4700 b.c. Thus the inhabitants of Mesopotamia must have had a workable elementary arithmetic. These same pioneers toward mathematics also invented or helped to transmit two major curses which continue to blight the unscientific mind, numerology (number mysticism) and astrol- ogy. It is an open question which of astrology or astronomy 26 THE AGE OF EMPIRICISM 27 preceded the other. Arithmetic of some sort necessarily came before numerology. For Egypt, the early historical record is somewhat more de- tailed. The more liberal of rival Egyptian chronologies assigns 4241 n.c. as the earliest precise date in history, this coinciding with the adoption of the Egyptian calendar of twelve thirty-day months with five days of feasting to complete the 365. This date also is supported by inconclusive astronomical evidence, correlat- ing the heliacal rising of the Dog Star Sothis, our Sirius, with the date at which the annual inundation by the Nile could be expected. Here again the impulse to develop astronomy, and hence also arithmetic, was agricultural necessity unless, of course, it was astrology. The geographical location of Sumer was more propitious than that of Egypt for a rapid development of the mathematics con- ceived in agriculture and born in astronomy. Egypt lay far off the main trade route between East and West. Sumer, the non-Semitic predecessor of Semitic Babylonia, lay directly across the path of the merchants at the north end of the Persian Gulf. Commerce stimulated mathematical invention in Sumer and ancient Mesopotamia as it probably never has since. Europe of the late Middle Ages also profited mathematically through trade; but the gain was in a diffusion of knowledge rather than in the creation of new mathematics necessitated by commerce. Possibly of greater importance than trade for the develop- ment of mathematics were the demands of primitive engineering. Both the Babylonians and the Egyptians were indefatigable builders and skilled irrigation engineers, and their extensive labors in these fields may have stimulated empirical calculation. But it would be gratuitous generosity to infer that because the Egyptians, say, succeeded in raising huge obelisks, they were therefore engineers in any sense that would now be recognized as scientific. Ten thousand slaves can muddle through the work of one head; and the apparent marvels of ancient engineering that impress us today may be only monuments to a lavish expenditure of brawn and a strict conservation of brains. The Israelites and others whom the Egyptians persuaded to take up practical engineering do not seem to have been greatly impressed by the technical skill of their overseers. Reliable evidence shows that arithmetic and mensuration in Babylonia developed from the early work of the non-Semitic Sumerians. This gifted people also invented a pictorial script 28 THE DEVELOPMENT OF MATHEMATICS which evolved into the efficient cuneiform characters that were to prove adequate for the expression of their arithmetic and mensu- ration. The political absorption of the Sumerians by physically but not intellectually more vigorous peoples occurred about 2000 b.c. Astronomy and arithmetic continued to flourish and, what is of singular significance, a sort of algebra evolved with incredible speed. This early appearance of algebra is one of the most re- markable phenomena in the history of mathematics. For all that is known to the contrary, other early civiliza- tions may have made progress toward mathematics comparable to that of Mesopotamia and Egypt, records of these two having survived largely by physical accident. The semiarid climate of Egypt and the inordinate reverence of the Egyptians for all their dead, including bulls, crocodiles, cats, and human beings, united to preserve the papyri that must have perished in a harsher atmosphere, and kept the memories of common things colorful for thousands of years on the walls of tombs and temples. Some of the most interesting historical documents as yet re- covered from the past survived only because the Egyptian morticians discovered that useless papyri made excellent stuffing to plump out the mummies of sacred crocodiles to lifelike obesity. The Babylonians impressed their records on a yet more durable medium, clay tablets, cylinders, and prisms, baked in the sun or in kilns. Sharpened sticks, one like an implement still used by children in modeling, indented the wedge-shaped charac- ters in the soft clay, and the baking fixed a record more durably than any printers’ ink on the toughest paper. Wars and the long decadence of a great civilization for once conspired to preserve some of the best in that civilization. The baked tablets, resistant to damp, rust, and pressure, and immune to the attacks of worms and insects, were buried beneath the mud ruins of dissolving temples and libraries. It would be easier for some science-hating zealot to obliterate modern mathematics than it would be for us to destroy the mathematical records of Babylonia. There is no reason to suppose that all the mathematical bricks have been exhumed. If the records themselves are solid and tangible beyond dis- pute, the like cannot be claimed for their interpretation. The reading of the most suggestive parts of the Sumerian and Baby- lonian records is a matter of great difficulty, demanding an un- usual combination of linguistic, historical, and mathematical TEE AGE OF EMPIRICISM 29 talents. Several points of interest are still in dispute among the scholars who since 1929 have finally broken the seals on ancient Babylonian mathematics. We shall not find it necessary to use any of this disputed material in order to give an idea, sufficient for our purposes, of what the Babylonians accomplished. What remains after the few doubtful items are discarded is impressive enough. Those far-off centuries in Babylon and Egypt are the first and last great age of the empiricism that led to mathematics. Above a multitude of details, five epochal landmarks survived for the guidance of later centuries. Number was subdued to the service of astronomy and commerce; the perception of form was clarified in an empirical mensuration and applied to astronomy, surveying, and engineering; the vast extensions of the natural number system which mathematics uses today were initiated; a method more powerful than arithmetic was begun in an algebra more than well begun; and last, also perhaps most significantly, practical difficulties in mensuration compelled some of those early empiricists to grapple at least subconsciously with the con- cept of the mathematical infinite. From that day to this, a stretch of nearly four thousand years, the struggle to compass the infinite has continued, and the record of the struggle is mathematical analysis. Possibly of greater significance for the future of the race than all the technical advance toward mathematics was another for which that advance was to be largely responsible. It dawned on the human mind that man might dispense •with the thousands of capricious deities created by human beings in the childhood of their race, and give a rational account of the physical universe. Although an explicit statement of this possibility was to be re- served for one of the earliest and greatest of the Greek mathe- maticians, it was anticipated by the astronomers and scientists of Egypt and Babylon, and it was there that our race began to grow up. Arithmetic- to 600 b.c. Since 1929 our knowledge of mathematics in ancient Baby- lonia has been increased many times over all that was previously known, largely through the pioneering work of O. Neugebauer (1S99-). Apart from their great intrinsic interest, these new accessions are extremely suggestive as possible clues to the THE AGE OF EMPIRICISM 31 the sexagesimal (60 as base), with a slight admixture of the decimal system with 10 as base. It has been conjectured that the 10 commemorates finger counting, while the 60 is 6 X 10, the 6 being adopted so that the useful fractions l/2 a 3 b 5 e ( a , b, c non-negative integers) would be expressible in finite terms. Traces of the sexagesimal system survive in our reckoning of time and in the corresponding divi- sion of the circumference of a circle into 6 X 60 degrees. But it is no longer universally supposed that such considerations in- duced the Sumerians to choose 60 as a base, and still less that the zodiac influenced their choice. The place-value system was used for both positive and nega- tive powers of the base. Thus, in the appropriate cuneiform symbols, 17, 35; 6, 1, 43, the semicolon indicating the beginning of the fractional part, denotes 17 X 60 + 35 -f- 6/60 -f 1/60 2 -f-43/60 3 . The method of writing sometimes introduces am- biguities; but the pertinacity of modern scholarship has removed these and made sense of the residue. According to the mood of the scribe, a blank might or might not indicate absence of the corresponding power of 60. This particular difficulty was over- come by the introduction of a special character for zero but not, probably, until the time of the Greeks. The great practical invention of zero has usually been attrib- uted to the Hindus, and it may still be debatable whether they or the Babylonians were first. If, as seems highly probable, the Babylonians were original in their invention, zero is an interest- ing example of the independent origins of mathematical ideas in different cultures. Zero also appeared in the arithmetic of another gifted people, the Mayas of Central America, who used 20 as a base and had a system of place-value. The Maya numeration has been assigned to the period a.d. 200-600. Their calcndric cycles go back to 3373 b.c., but this does not imply that the Mayas were civilized or even in existence that early. The Babylonians were among the most indefatigable com- pilers of arithmetical tables in history. Since it was easier to multiply than to divide, they tabulated 1/n for integers n adapted to the base 60. Other reciprocals, ‘irregulars,’ like i, -Jy, naturally caused more trouble, but were competently avoided by manufacturing problems in which such awkward divisors would automatically drop out in the course of the work. This is not the only instance in Babylonian mathematics where the teacher or the pupil appears to have applied a technique once 32 THE DEVELOPMENT OF MATHEMATICS classical in mathematical physics: given the solution, find the problem. There were also multiplication tables for such multi- pliers as 7, 10, 12^, 16, 24, etc. Tables of squares appropriately read served as tables of square roots, and likewise for cubes. Another table listed values of n 3 + n 2 for n = 1, 2, . . . , 30. The peculiar significance of this strange tabulation -will appear when we come to Babylonian algebra. From all this and a great deal more of a similar character, it is evident that the Babylonians of about 2000 b.c. were highly skilled calculators. It may not be too generous to credit them with an instinct for functionality; for a function has been suc- cinctly defined as a table or a correspondence. Historically, the most remarkable thing about this rapid progress in the subjugation of number is that it appears to have been ignored by the Greeks of the sixth century b.c. For what now seems to us the simplest, most natural development of mathematics this was a calamity. The fact that it happened casts a slight shadow of doubt on the vaunted intelligence of the early Greek mind. But since to press this point would be tanta- mount to historical blasphemy, we merely suggest that the mathematically informed observer examine the evidence and reach his own conclusions, even at the risk of upsetting sacro- sanct tradition. Egyptian arithmetic shows even more starkly its laboriously empirical origin. As early as 3500 b.c., the Egyptians freely handled numbers in the hundreds of thousands. Their hiero- glyphics of this early date actually record the capture of 120,000 human prisoners, 400,000 oxen, and 1,422,000 goats. The last is probably a poetic flight of the conqueror’s imagination — the catalogue occurs on a royal mace — for even today the experts of the U.S. Census Bureau would be exercised to enumerate that many goats in the brief interval between victory and celebra- tion. But the enthusiastic exaggeration, like that of the ancient Hindus in multiplying their gcds practically to infinity, shows at least that the Egyptians of 3500 b.c. had completely overcome the inability of primitive peoples to think boldly in terms of numbers. The significance of this advance can be appreciated only by comparison with the arithmetical backwardness of peoples well beyond barbarism today and also, as will appear, once more with the Greeks. '• Egyptian numeration followed the decimal system, but with- out place-values. The arithmetic of about 1650 b.c. was capable THE AGE OF EMPIRICISM 33 of addition, subtraction, multiplication, and division, and was applied to numerous extremely simple problems involving all these operations. In fractions, 3 was denoted by a special symbol; other fractions were reduced to sums of fractions of the form l/«, n an integer. In the Rhind papyrus of about 1650 b.c., copied by the scribe Ahmes (A’h-mose) from an older work, divi- sions are performed by means of these ‘unit fractions,’ the technique being the expression of m/n, in > 1 , as a sum of unit fractions; for example, + Trhr + rW- How such curi- ous resolutions were first obtained seems not to be known. They may represent the fossil experience of centuries carefully pre- served in tables for future use, as we today store up logarithms. Ahmes transmitted resolutions of all fractions 2 /«, where n is any odd number from 5 to 101. These could have been derived from successive applications of the solution in positive integers of the so-called optical equation, l/x + 1 /y = l/n, but it is most improbable that they were. Of the many other conjectures, none is acceptable to a majority of competent scholars. All of the problems solved arc childishly simple. Some arc quite delightful for their unintentional revelations of Egyptian manners and customs, as when Ahmes goes into the arithmetic of bartering beer for bread and vice versa. Either the Egyptians were less puritanical in their schools than we, or Ahmes intended his treatise only for expert mathematicians. Less inflammatory problems arc concerned with the rationing of oxen and various kinds of birds from geese to cranes and quail. A more fanciful kind, obviously of no earthly use to anyone, recalls the older type of English examination question and the more antique efforts of our College Entrance Examination Board. Loaves of bread arc partitioned among several imaginary beings who are to receive amounts in arithmetical progression. There is nothing new, provided it be silly enough, under the sun. The most significant detail for the development of mathe- matical thought in all the Egyptian arithmetic is the occasional checking of a calculation. This seems to show that the Egyptians of at latest the seventeenth century b.c. understood the value of proof in arithmetic. If this is a justifiable conclusion, those ancients were well on their way to mathematics 'when, un- accountably, they stopped. It is said that the Egyptians’ arithmetic was sufficiently advanced for the simple demands of their daily affairs. Of greatet interest, perhaps, for the evolution of mathematics are those t'ery TTIE AGE OF EMPIRICISM 35 century B.c. of the correct rules of signs in multiplication. How the Greeks overlooked all this is a mystery. Algebra without symbolism - Passing on to Babylonian algebra of about 2000 b.c., we come to what historians consider the most remarkable anticipation in the development of mathematics. First is the question of proof, without which mathematics in the accepted sense does not exist. Did the Babylonian successors of the Sumerians have any conception of deductive reasoning? No categorical answer can be given. So far (1945) there has not been discovered any Babylonian record of a mathematical dem- onstration. But this is not necessarily conclusive against at least a mute intuition for proof, evidence for which is overwhelming. To put the case as favorably as seems justified by the undis- puted evidence, we may picture a teacher of elementary algebra today grading an examination on quadratic equations. The pupils have been asked to solve 12* 2 — 7* = — 1. Some h ave substituted in the standard formula x = (— b ± \b 2 — 4ac)/2a for the solution of ax- -f- bx + c = 0 and have been satisfied with one root; others have ‘completed the square’; and one original genius has ‘normalized’ the equation by multiplying throughout by the coefficient of x 2 , getting (12 *) 2 - 7(12*) = -12, before solving for 12*, whence he easily finds * by division. Falling short of the better Egyptians 3,500 years before him, not one of the pupils has sought to verify his solution by sub- stituting in the equation, as the harried teacher had forgotten to demand a check. Nor has anyone offered a word of proof in support of his formal calculations. All have gone through the steps to a solution as if the teacher, with an open book in her hand, were dictating from first to last what to do next. Equivalents of all this, including the absence of verification, occur in the Babylonian tablets of about 2000 b.c. Verbal instruc- tions direct the solver to follow a path leading to the solution by our standard formula, or to normalize the equation, or to complete the square. It is algebra by rule and without algebraic symbolism. The scribes who indented these paradigms on the soft clay, or who directed others, certainly had general pro- cedures in mind. But it must be admitted that correct general 36 THE DEVELOPMENT OF MATHEMATICS rules, even when successfully applied to hundreds of special examples, do not constitute mathematical proof. Ail of the Babylonian algebra is of this character: detailed solution of one numerical problem after another by verbal instructions follow- ing definite patterns. No pattern is ever isolated as a general procedure. The empirical character of the algebra of the Babylonians, and perhaps also their social outlook, is even more strikingly evi- dent in their astonishing solutions of cubic equations with numerical coefficients. Expressed in our terminology, equations of the type x z -f- px 2 -f- q = 0 are reduced to the normal form y 3 -j- y 2 — vrith y as x/p, t = —q/p z , by multiplying the original equation throughout by 1/p 3 . If the resulting r is posi- tive, the value of y, and hence that of x, is obtainable from tabulated values of n z -f- n 2 , provided r is in the table. From the equations actually solved in this manner, it is con- ceivable that the scribe proceeded from certain tabulated ris to construct his equations x z -f- px 2 = q so that they would be solvable. Then he triumphantly produced the solution. If so, his pupils must have been as thoroughly mystified as is any student vrith a grain of mathematical intelligence today when his mys- terious instructor pulls mathematical rabbits out of invisible hats. Brilliant trickery is no longer considered reputable mathe- matics. But if it is true, as has been asserted, that the mathe- matics of Babylonia and Egypt was the jealously guarded secret of a priestly sect, the mystery vanishes. One of the greatest services the Greek mathematicians rendered civilization was their shattering of the tradition of secrecy fostered by self-per- petuating priesthoods. The attempt of Pythagoras to carry on the secretive tradition of Babylon and Egypt was quickly dis- sipated, and enlightenment was put within the grasp of any unsanctified vulgarian with the will and the intelligence to reach for it. It has been conjectured that the reduction of the general cubic to the above normal form was within the powers of the Babylonian algebraists. But we need not assume this much to grant that the Babylonians had taken a long stride toward mathematics in what they actually did. For the spirit animating mathematical discovery is the recognition of uniformity in a host of apparently diverse phenomena. The project of reducing a multitude of particular equations to a standard form, or even the easier inverse problem of constructing special equations THE AGE OF EMPIRICISM 37 indefinitely to fit a prescribed solution, would occur only to an intelligence that was essentially mathematical. This methodology of transformation and reduction, general- ized and many times refined with advancing knowledge, runs like a scarlet thread through all the greater epochs of mathematics. A relatively difficult problem is reduced by reversible transfor- mations to a more easily approachable one; the solution of the latter then drags along with it the solution of the former and of all problems of which it is the type. The Babylonian reduction of cubics appears to be the first recorded instance of this method- ology. We shall observe it again in the Italian algebra of the early sixteenth century, and in Vieta's signal advance half a century later. In geometry, to cite but one instance, the method first appears in the device of central projection, whereby geom- eters in the seventeenth century derived the properties of conics from those of the circle. Taking the view, as we shall in general, that uniform methods are of more lasting significance than the sum total of the indi- vidual results, however brilliant or useful, obtained up to any given epoch by their use, we might rest the case for the Babylo- nians as mathematicians on their reduction of cubics to a normal form. But the spectacular ingenuity of their algebra — when we consider that nothing surpassing it was known in Europe till the sixteenth century a.d. — demands a summary indication of certain particulars. Always relying on their extensive numerical tables, the Baby- lonian algebraists solved simultaneous linear equations in two unknowns, also simultaneous quadratics of the type xy *= 600, (fl.v + iy) 2 + cx -f- dy — e for 55 sets of special numerical values of a, b, c, d, c , each of the sets leading to a quadratic in x. They also proposed a problem leading to the general quartic which, it need hardly be said, they did not solve; and likewise for a general cubic arising from a problem on frustums of pyramids. A cubic in x 1 also appears. In their solutions of quadratics the Babylonians were usually content with one root, although in one example both roots (positive) are given. A multiplicity of unknowns does not seem to have dismayed them, one problem leading to ten linear equations in ten unknowns. Even more remarkable, perhaps, is the successful solution by initial trial and subsequent interpolation of an exponential equation to determine the time required for a sum of money to double itself at a stated rate of compound interest. Such equa- 38 THE DEVELOPMENT OF MATHEMATICS tions are solved today by logarithms. But to infer that the Babylonians understood logarithms even to base 2 would be as fantastic as that classic fable of archaeology which declares that the ancient Egyptians were familiar with wireless te- legraphy because not a scrap of wire has been found in their tombs. In another direction, the Babylonians partly anticipated the summation of a geometric progression by Archimedes in the third century b.c., giving the correct result for ten terms by a special case of the general rule. Of greater significance for the future of mathematics was the highly intelligent reaction of the Babylonian algebraists to irrationals. Their tables and their equations taught them that not every rational number in their tables had a tabulated square root. Faced with this fundamental fact, they proceeded to approximate by means of the rules ( a 2 + b 2 )* = a -f- & 2 / 2 i 2 , or = a 2 -\- lab 2 . The first is reasonable and reappears about two thousand years later with Heron of Alexandria; the second is hopelessly wrong, being dimensionally impossible. We note in passing that the reasonable approximation is obtainable from Newton’s binomial series; but again this does not imply anticipa- tion. In further approximations to quadratic surds they used what may be interpreted as the first steps toward conversion into periodic continued fractions. For they gave the approxi- mation ly^-, correct to two decimals. As will be seen when we consider the Pythagoreans, yj2 marks one of the cardinal turn- ing points in the history of mathematics. Contemplating work of this caliber, done for the most part about two thousand years before the Christian era, we can only marvel how it was done, for we do not know. The detailed numerical solution of specific examples gives no hint of the thought inspiring the uniform procedures. Neugebauer empha- sizes that the technique is based on elaborate numerical tables. At the lowest estimate, the high skill in using such tables indi- cates an extraordinary capacity for detecting uniformities in masses of empirical data. The Babylonians were the world’s first exact astronomers; and so accurate were their first observa- tions and their calculations that Kidinnu, about 340 b.c., made the capital discovery of the precession of the equinoxes, antici- pating Hipparchus by about two centuries. It seems reasonable to assume that unrecorded centuries of observing the planets led to the accumulation of numerical data from which a purely THE AGE OF EMPIRICISM 39 rhetorical algebra evolved. For Babylonian algebra -was entirely unsymbolic. Alore remarkable still, the processes which now would be summarized in formulas were never, so far as is known, reduced to written rules. If these elaborate procedures were transmitted wholly by word of mouth, the strain on even a strong memory must have been considerable. Another unsolved problem is even more puzzling. Up to 1900 it was customary to ascribe the beginnings of algebra to the Greek Diophantus in the third century a.d., over two thousand years after the Babylonians had bettered some of his best. Where was algebra buried in the meantime? It has been con- jectured that the Greeks of the sixth and fifth centuries b.c. must have been acquainted with what the Babylonians had done in algebra because, as will appear, a considerable amount of Baby- lonian empirical geometry almost certainly was known to those same Greeks, Direct evidence is lacking that the early Greeks were not acquainted with Babylonian algebra, but the indirect evidence is at least worth noting. For if the early Greeks were cognizant of Babylonian algebra, they made no attempt to develop or even to use it, and thereby they stand convicted of the supreme stupidity in the history of mathematics. But it is commonly agreed that the early Greek mathematicians and philosophers were among the most intelligent human beings that ever lived. This awkward historical dilemma can be circumvented by a slight anticipation. The ancient Babylonians had a rare capacity for numerical calculation; the majority of the Greeks were either mystical or obtuse in their first approach to number. What the Greeks lacked in number, the Babylonians lacked in logic and geometry, and where the Babylonians fell short, the Greeks excelled. Only in the modern mathematical mind of the seven- teenth and succeeding centuries were number and form first clearly perceived as different aspects of one mathematics. Nothing has been said about Egyptian algebra because it was far less advanced than the (probably) earlier work of the Baby- lonians. Between 1850 and 1650 b.c. the Egyptians solved easy numerical equations of the first degree by trial, or by what was called the rule of false position in the Middle Ages. The last makes it plausible that the Egyptians understood proportion. If they did, and historical experts do not doubt it, they share with the Babylonians the honor of having uncovered a main root of mathematical analysis. 40 THE DEVELOPMENT OF MATHEMATICS Toward geometry and analysis 3 Babylonian mensuration 2 of about 2200-2200 b.c. is almost as astonishing as the contemporary algebra. Mathematically, it is of the same character as the algebra in its disregard of proof. Correct rules are applied for finding the area of any rectangle, right triangle, isosceles triangle, trapezoid with one side per- pendicular to the base, and “if t be taken equal to 3,” any circle. This approximation to 7 r is famous also for its occurrence in the Old Testament. A little later, between 1850 and 1650 b.c., the Egyptians had the closer approximation ~ 3.16. It would be interesting to know what suggested the curious (g-) 4 . In their mensuration of solids the Babylonians of about 2000 b.c. gave correct solutions of numerical problems involving rectangular parallelepipeds, right circular cylinders, and right prisms with trapezoidal bases. Some of this had obvious applica- tions to earthwork problems in the excavation of canals for drainage or irrigation. Their rule for the volume of a truncated square pyramid was incorrect. The correct rule of the Egyptians, one of the most remarkable achievements of pre-Greek mathe- matics, will be considered later by itself. Passing to theorems of pure geometry known to the Baby- lonians of the same period, we select three for their outstanding historical suggestiveness. The first two are: the angle in a semi- circle is a right angle; the Pythagorean theorem c 2 = a 2 + b 2 t where c,a,b are the sides of a right triangle, for certain numerical values of c,a,b, as 20, 16, 12 and 17, 15, 8. The first of these, often considered one of the most beautiful theorems in elementary geometry, is said to have been proved by the Greek Thales about 600 b.c. It would be guessed immediately on inscribing rectangles in circles. The Babylonians offered no justification. From the second, and certain numerical calculations for the sides of right triangles, it has been argued that the Babylonians knew the Pythagorean theorem in the general case, but the evidence seems inconclusive. Until 1923 it was supposed that the Egyp- tians knew this theorem at least in the case 5 2 = 4 2 + 3 2 , be- cause the Egyptian ‘rope stretchers’ were.formerly said to have used this property of 5, 4, 3 in laying out right angles for the orientation of buildings. But it is now claimed that even though 5, 4, 3 may have been used thus, the Egyptians knew not a single instance of the Pythagorean c 2 = a 2 -j- b 2 , because there is no documentary evidence that they did. Since c, a, b are the sides THE AGE OF EMPIRICISM 41 of a right triangle if and only if c 2 = a- + b 2 , we have here an interesting historical puzzle as to how the Egyptians guessed what they needed. Regarding the Pythagorean theorem itself, whoever first guessed it, we recall that it is the cornerstone of Euclidean metric geometry and one of the bases of all metrics. It too, like similar triangles, threads all mathematical history, not only in geometry, but also in algebra, the theory of numbers, and mathe- matical physics. The third significant empirical theorem of the Babylonians in pure geometry is the earliest recorded trace of the origins of mathematical analysis: the sides about corresponding angles of similar triangles are proportional. This theorem implies equality of ratios. It has been said to follow that the Babylonians had some conception of ratio. But, if we wish to be as precise here as we were a moment ago in the case of the Pythagorean theorem versus the nonsuited Egyptians, we have no right to assert that the Babylonians actually had even the remotest conception of ratio. For 'equal ratios’ and ‘ratio’ are distinct concepts in mathematics, and an extensive theory of ‘equal ratios’ is easily possible without any definition of ‘ratio.’ ‘Ratio’ is on a higher level of abstraction than ‘equal ratios.’ Euclid attempted, not too successfully, to define ratio. His definition has been trans- lated by De Morgan thus: “ratio is a certain mutual habitude of two magnitudes of the same kind depending upon their quantu- plicity.” Fortunately Euclid never had to appeal to this abstruse definition, his ‘theory of ratio’ being wholly a theory of propor- tion, that is, of equal ratios. That the Babylonians, or anyone else before the nineteenth century, had a workable conception of ratio seems extremely improbable. The ratio of in to n, written usually as m/n, is understandable only — so far as we know at present — as a number-couple (?«, n) with certain postulated properties for the four rational operations on such couples. So far as documentary evidence goes, there is none, apparently, to show that the Babylonians ever got within hailing distance of Euclid, who, if he did not succeed in giving an unmystical defini- tion of ratio, at least acted as if he were aware that ‘ratio’ and ‘proportion’ are different concepts. But in this whole matter we have no desire to be as precise as we were in the case of the Egyptians; we have tried merely to indicate where the mathe- matical crux of the history lies. With their numerical examples of four numbers in proportion, the Babylonians took the first 42 THE DEVELOPMENT OF MATHEMATICS step toward the Greek theory of proportion which has lasted, practically unmodified, to this day. Another possible source of much modern mathematics will be noted presently in connection with the pyramid. But the sub- sequent history of what evolved from similar triangles is so clear, and of such outstanding importance for all mathematics, that we shall leave Babylonian metrics here with this as its crown. With one exception to be discussed presently, the Egyptian empirical mensuration is less impressive than the Babylonian. From their prodigious architecture it might be reasonable to infer that the ancient Egyptians were skilled construction engi- neers and hence at least respectable geometers. They were neither. Brute force in the form of unlimited slave labor made brains all but superfluous. Until it was discovered how they raised the huge stone blocks to build their pyramids, it was supposed that the Egyptian overseers were acquainted with at least the rudiments of scientific engineering. What they actually did 4 puts them on the intellectual level of the ants. As the suc- cessive tiers of a pyramid rose, the slaves laboriously buried under thousands of tons of sand the face of the work already done. The swarms of slaves lugged the blocks up the long ramp. When the task of building was finished, the slaves removed the mountain of sand burying the pyramid, to put it all back where they had got it in the first place. The dazzling result of their labors shone out in all its splendor, another time-outlasting monument to the unconquerable spirit of man’s temporal rulers and the unbreakable backs of those who do the work. The sagac- ity of the slave-driving Egyptian overseers has been rated highly by competent enthusiasts. In accuracy of practical measurements both the stonemasons and the irrigation engineers of Egypt of the third millennium n.c. reached great heights. It is asserted, for instance, that the maximum error in a side and in a corner angle of the Great Pyramid are only small fractions of one per cent. Again, the sur- veyors responsible for observing the Nile succeeded in placing their water gauges in one plane for a distance of about 700 miles round all the bends of the river. With a sufficient number of centuries for observation, this could be done by trial and error, and it docs not necessarily imply any great knowledge of scien- tific surveying. The Egyptians had plenty of time. In the direction of geometry they seem to have known that the area of any triangle is obtainable by the rule •J- base X aid" THE AGE OF EMPIRICISM 43 tude. They also computed the volume of a cylindrical granary correctly. These results are as advanced as anything the Egyp- tians are known definitely to have obtained, with the exception of their work on the pyramid to be noted next. For a people who achieved magnificent art, it must be admitted that the Egyp- tians’ efforts toward geometry are mostly trivial and disappoint- ing. This, probably, is only to be expected, as acceptable art is created by peoples but little above savagery. The greatest Egyptian pyramid Every list of the seven wonders of the ancient world includes the Great Pyramid. But since the translation of the Moscow papyrus 6 in a.d. 1930, this pyramid has been overtopped by a greater than any the slaves in Egypt could ever have reared. This greatest of Egypt’s pyramids existed only in the mind of a nameless mathematician who discovered or guessed the most remarkable result in pre-Greek geometry. He gave a numerical example of the correct formula, -g-A(fl 2 -f- ab + b 2 ), for the volume of the frustum of a truncated square pyramid, h being the altitude and a,b the sides of the top and bottom bases. This numerical application of a special case of the prismoidal formula dates about 1850 n.c. It is not known how this formula was obtained. Of several plausible conjectures, none is accepted by a majority of reconstructive scholars. Had the forgotten Egyptian responsible for this result proved his procedure, he would rank high among the greater creators of mathematics. Even the empirical discover}’- of such a process or its verbal equivalent is evidence of extraordinary mathematical insight. In some guise the essential method underlying the formula has reappeared in all the great ages of mathematics. The Greeks called it exhaustion; 6 Cavalieri in the seventeenth century called it the method of indivisibles and, as will appear in the proper place, got no closer to proof than the ancient Egyptians of at latest 1850 n.c. To us it is the theory of limits and, later, the integral calculus. The reasons for believing that no Egyptian could ever have even distantly approached a proof recur many times in mathematical history; the final and con- clusive one was stated only in a.d. 1900. The complete method of exhaustion is sufficiently described through the simpler problem of determining the area of a circle. Regular polygons of n sides arc inscribed and circumscribed to the circle; the required area is less than that of the circum- 44 THE DEVELOPMENT OF MATHEMATICS scribed polygon and greater than that of the inscribed; as n is increased, the difference between the areas of the polygons diminishes until, in the limit, as n tends to infinity, the differ- ence vanishes, or is ‘exhausted,’ and the common area of the limiting polygons is equal to that of the circle. In many partial applications of the method, only inscribed polygons were con- sidered. In either variant, it is necessary to know the area of a regular polygon of n sides. This is immediate once the area of an isosceles triangle is known. If the limits described exist, and if they can be calculated, the problem is solved. At any stage, say n = 96, where Archimedes stopped in the third century b.c., an approximation to the area of the circle is obtained from the calculable polygons. Moreover, this approxi- mation is comprised between determinate bounds given by the areas of the inscribed and circumscribed polygons of 96 sides. But the crucial step in obtaining the exact formula for the area, or even defining what is meant by the area , is taken only by passage to the limit as n becomes indefinitely great. For the truncated square pyramid we might proceed similarly by inscribing and circumscribing stairways whose steps are rec- tangular prisms with square bases; and it is conceivable that the Egyptian inferred his rule from the easily calculated approxima- tions given by stairways with a few steps. Indeed, the earlier pyramids were of this type, and the Great Pyramid itself pre- sented just such an appearance before the final smooth sheathing of dressed stone was applied. But however the Egyptian reached his rule, his intuition gave him the correct result that is provable only by the integral calculus in some guise. For all proofs of the prismoidal formula and its special cases ultimately appeal to the formula for the volume of a triangular pyramid. The trivial generalization of this result for a pyramid on any polygonal base was attributed by Archimedes to Democritus, the founder of atomism, in the fifth century b.c. Criticism by the Greek sophists of the limiting processes used by Democritus and others was partly responsible for the particular course which mathematics followed in ancient Greece; and this was one of the major turning points in the evolution of mathematics. It would be interesting to know whether Democritus was influenced by the Egyptian result. He was one of the most widely traveled and more boastful of the early Greeks, bragging that although the Egyptian rope stretchers taught him all they knew, he himself knew far more. Might it not be possible, it may be asked, that the Egyptian THE AGE OF EMPIRICISM 45 obtained his rule by some device obviating the theorem of Democritus ? This would be so only if it were possible to prove by elementary means that triangular pyramids of equal altitudes are to one another as their bases. Euclid’s proof in his Elements is not elementary in that it is by the method of exhaustion, implying the concept of continuity. It was this concept to which the sophists objected so pertinently that they succeeded in de- flecting mathematics into a narrow channel which, to a modern mathematician, seems forced and unnatural. The consideration of this will be our principal concern when we follow mathematical thought through ancient Greece. Conceivably, a strictly finite proof of Euclid’s basic theorem for triangular pyramids might be possible. The lack of such a proof might be due only to mathematical incapacity and not to the nature of mathematics. Were such a proof possible, the Egyptian might well have proved his rule, or at least have perceived, however dimly, some mathematical grounds for it. Recognizing the fundamental mathematical significance of the possibility of a strictly finite proof for Euclid’s theorem, C. F. Gauss (a.d. 1777-1855, German), usually rated with Archi- medes (287 ?— 212 b.c., Greek) and Isaac Newton (a.d. 1642- 1727, English) as one of the three greatest mathematicians in history, in 1844 urged that a proof not depending upon con- tinuity be sought. Thus it was by no means obvious to Gauss that such a proof might not be found. In 1900, M. W. Dehn proved that no such proof is possible. It seems unlikely, then, that the Egyptian had anything resembling a proof for his rule. If it was just a lucky guess, he was so good at guessing that he needed no mathematics. Several of the greater mathematicians have emphasized intuition in mathe- matics as the necessary spark without which there is no discovery. Some have even discounted proof almost to zero, claiming that any competent hack can grind out a proof once the result has been guessed. Measured by this demotic standard, the nameless Egyptian was a very great mathematician indeed. The contribution of Babylon and Egypt It has been said that no subject loses more when divorced from its history than mathematics. This may be true, but there is a sort of converse which is equally true. The history of no sub- ject loses more when divorced from its subject than does the 46 THE DEVELOPMENT OF MATHEMATICS history of mathematics. With this in mind, we recall that we are primarily interested in the development of mathematical thought, rather than in the exhibits in a museum of antiquities. It is now time to apply this primary interest to our first collection of specimens. With the memorable achievements of Babylon and Egypt behind us, let us glance back for a moment, forget all the human struggling that made these long-dead things once live, and estimate them solely in the light of mathematics. One item of all the treasure will suffice as typical of all. The Babylo- nian mensuration of the circle throws into sharp relief the distinc- tion between what is mathematics and what merely resembles mathematics. In the familiar formulas lirr, 7 rr 2 for the circumference and area of a circle of radius r, 7 r denotes a constant number which, to seven decimals, is 3.1415926. The last of course is of great practical importance. But the long chronicle of 7r signifies vastly more for the history of mathematics than a rather dreary record of successive approximations from the crude 3 of the Babylo- nians of about 2000 Rtc. to the 707 decimal places, all but a few of them quite useless, of W. Shanks in a.d. 1853. Any tyro in geometry understands what is intended by such an elliptical statement as “the ancient Babylonians took7T equal to 3.” But accepted literally, this statement implicitly denies the existence of mathematics and makes nonsense of its history. So far as is known, nobody before the ancient Greek mathemati- cians ever “took7T equal to” anything. Until it had been proved that the ratio of the circumference of any circle to its radius is in- dependent of the radius, or that (Euclid, XII, 2) the areas of any two circles are to one another as the squares on their diameters, there was no “ 7 r” to be taken. Induction from physically measured circles may have sug- gested to some empiricist that the circumference of any circle is greater than 3\\ diameters and less than 34, the bounds proved by Archimedes to exist. But only a mind very immature scientifically would trust these bounds for circles either so small or so large that they could not be measured by the means used for the others. Certainly no mind with the faintest stirrings of a mathematical instinct would trust them. Induction from practi- cal experience is not enough here; mathematics is demanded. If, in this particular matter of t, it be argued that the ancients before Greece had no need for mathematical — not merely numerical — precision, and that close induction from THE AGE OF EMPIRICISM 47 experience sufficed, several replies may be given, all pertinent and applicable to the entire history of mathematics. First, on the severely practical side, any civilized people using a calendar would need sooner or later to know, or at least to believe, that there is a constant, say c, such that the circumference of a circle is c times its radius. Otherwise, as their astronomy became more exact, they would live in constant dread that their calendar might begin fluctuating disastrously, and with it, their commerce and agriculture. Second, mathematical precision and numerical precision are very different things, in spite of what some practical souls may imagine to the contrary. A fair degree of numerical precision was demanded in ancient times. Had civilization crystallized in the second millennium before our era, no greater precision in nu- merical calculations than that which sufficed in Babylon would have been required. But, to cite only three instances of the need for greater numerical precision as civilization evolved, the calendar, geography, and navigation demanded an increasingly precise astronomy, and this was forthcoming only when arith- metic and geometry had progressed far beyond the sharpest exactness possible to mathematical empiricism. The validation of a formula is its proof, without which precision even in the narrowest sense of practical utility is impossible once the earlier stages of civilization are passed. A third distinction which sharply separates the Archimedean mensuration of the circle from the Babylonian is exactly the dis- tinction between scientific and prescientific thinking. A mind which rests content with a collection of facts is no scientific mind. The formulas in a mathematical handbook are no more mathe- matics than arc the words in a dictionary a literary masterpiece. Until some unifying principle is conceived by which an amor- phous mass of details can be given structure, neither science nor mathematics has begun. The first and most extensive of all the structures unifying number and form is deductive reasoning. There is no conclusive evidence that such reasoning was used in mathematics before the Greeks. They also advanced far beyond mythology in thcii attempts to unify r their observations of nature. Their cosmic speculations may have been too naive to be of much scientific value; nevertheless, they were deliberate steps away from mythology and superstition and toward science. With the con- scious recognition that unity and generality are desirable values 48 THE DEVELOPMENT OF MATHEMATICS both practically and aesthetically, mathematics and science became possible. All this may sound rather dogmatic, but it is not so intended. It is merely one of two possible points of view; and the reader is recommended to take the opposite side, follow it out consist- ently, and observe to what conclusions he is led in his estimates of mathematics and its history. The like applies to our entire future course, and in particular to the following paragraph, with which no doubt many will disagree. It is the estimate 7 of pre- Greek mathematics to which our argument has led us. Until, if ever, evidence is uncovered proving that the Greeks were anticipated in their conception of mathematics as a deduc- tive science, the greatest contribution of the Babylonians and Egyptians must remain their unconscious part in helping to make possible the golden ages of Eudoxus and Archimedes. It was enough, and should preserve their memory as long as mathe- matics lasts. 50 THE DEVELOPMENT OF MATHEMATICS For their percussions on the higher arithmetic of Fermat and others in the eighteenth to the twentieth centuries, the figurate numbers of the Pythagoreans (sixth to fifth centuries b.c.) may be remembered as one of the most suggestive contributions of arithmetica to the modern higher arithmetic. These numbers also achieved a certain prestige in Plato’s science, as for example in his Timaeus. The triangular numbers in particular, when insinuated into the Empedoclean chemistry of the four ‘elements,’ earth, air, fire, and water, were partly responsible for the remarkable metaphysical conclusion that “all matter is essentially triangles.” The figurate numbers are supposed to have originated from the representation of the regular polygons by placing a pebble on each vertex and then bordering the polygons in such a manner that regularity and number of sides were preserved. This possible origin has been instanced as an early occurrence of the connections between number and space. Whether this superficial connection is more than a mathe- matical pun, the square numbers pebbled out as described may be responsible for the persistence of ‘squares’ in our algebra, where geometric imagery is not only obsolescent but irrelevant. Another numerical item that might be credited to arith- metica as practiced by the Pythagoreans is the law of musical intervals, traditionally attributed to Pythagoras himself. The law relates the pitches of notes emitted by plucked strings of the same kind, under equal tensions, to the lengths of the strings. This discovery, the first in mathematical physics, revealed an unexpected interdependence of number, space, and harmony. It is scarcely surprising, then, that it precipitated a deluge of number mysticism. Human credulity being what it is, the resulting crop of esoteric philosophies and bizarre creeds which sprang up in ancient times, and which continue to flourish in our own, might have been anticipated. The Pythagorean law was also responsible for the retention of ‘music’ in the standard medieval curriculum. In fact, nearly every conceivable use, except a sensible one from a modern point of view, was made of the epochal discovery that musical sounds and numbers are related. The fact that had been discovered by experiment became the occasion for abandoning experiment in favor of the unaided human reason. Consequently, the experiment that might have started a scientific age, in the modern sense, aided most effectively in retarding that age for about 2,000 years. In logistic — computation — the Greeks did nothing that is FIRMLY ESTABLISHED 51 not best forgotten as quickly as possible by a mathematician. Their best attempt to symbolize numbers was a chiidish scheme little better than juxtaposition of the initial letters of number names. Yet the development of Greek numeration, such as it v.'as, might legitimately merit a great expenditure of erudition, time, and space in the antiquarian history of mathematics. Its interest here is negligible because, fortunately for mathematics, Greek numeration quickly perished. Only one of its many dis- abilities was its incapacity to represent even moderately large numbers concisely. Archimedes in the third century b.c. over- came this in a scheme of counting by eighth powers of ten. But as he just missed the place-system of numeration, his ingenious idea also perished. It is supposed that the Greeks themselves, except a few experts, made little use, if any, of their alphabetic numbers in computation, but resorted to the abacus. Sporadic attempts to rehabilitate the battered reputation of Greek logistic as a work- able system appear to be based on misapprehensions of what the Greeks actually did; and the majority opinion remains that of the conservative and sympathetic historian of Greek mathe- matics who characterized Greek numeration as vile. Some good thing undoubtedly came out of Nazareth, but it seems unlikely that any decent arithmetic could have issued from an inherently vile way of symbolizing numbers, and those who have gone most deeply into the matter assert that none did. 52 THE DEVELOPMENT OF MATHEMATICS Computation begins only after the mathematics has been done. Neither the Hindu numerals nor any others are of any importance whatever in vast tracts of modern mathematics. No numerical computations are performed. Gauss is reported to have lamented that his ancient compeer Archimedes just failed to anticipate the Hindu system of nu- meration. With his own prodigious astronomical calculations behind him, Gauss speculated how much farther advanced the science of the nineteenth century might have been had Archi- medes succeeded. If this report is accurate, it admirably points the parting of the ways. For Gauss had computational astronomy in mind; and it was Gauss the expert calculator, not Gauss the creative mathematician, who was lamenting the failure of Archimedes to take that last simple but essential step. Ex Oriente lux In an older day the sudden rise to maturity of Greek mathe- matics was classed with the miracles. Before the twentieth-cen- tury research on the records of Babylon and Egypt, it appeared that mathematics in Greece had grown from conception to vigorous manhood in a mere flash of about three centuries. Today we know that the respect which Greek writers themselves expressed for the wisdom of the East, even while extolling their own, was justified. The sudden maturity is no longer incredible. Modern science, beginning with Galileo and Newton, has developed with equal rapidity from origins relatively no more promising than those from which Greek mathematics evolved. The route by which the learning of the East reached Greece has yet to be uncovered in detail. But the battles of Marathon (490 b.c.), Thermopylae, and Salamis (480 b.c.), where the Greeks broke the Persians on land and sea, may have been a turning point in mathematical history as they were in that of all Western civilization. Those battles prove at least that young Greece was in close contact with ancient Persia, the imperial successor of Egypt, Babylonia, Phoenicia, Syria, and all Asia Minor. Marathon and Salamis are universally acclaimed by partisan humanists as unmixed benefits for the development of civilized culture. The career of mathematics hints that they may have been the beginning of a long detour round the origins of much that today is more vital than some of the Greek masterpieces. Until the route from East to West is traced, we shall not FIRMLY ESTABLISHED 5J know definitely how much Greek mathematics owed to its prede- cessors. Without in any way belittling the Greek contribution, we may safely believe that the extreme miracle of spontaneous generation did not happen in Greece. In this connection, two opposing theories of the anthropologists may be mentioned. Ac- cording to the first, closely similar cultures will evolve spon- taneously in similar environments, no matter how -widely separated. According to the second, all civilization is propagated from foci of culture, which in turn were civilized from more remote foci, and so on, until all culture is traced to one initial focus, usually Egypt. A third theory combines the patent advantages of both. The spontaneous theory is echoed in the frequent remark that mathe- matical and scientific discoveries are often made independently and almost simultaneously by two or three men. On the diffusion theory this can be explained by observing — -what is the fact — that in such instances the discoveries usually germinate in a body of knowledge accessible to all. Something roughly like this is now believed to have been responsible for the sudden efflorescence of Greek mathematics. The lore of the East was available to any curious Greek who could afford a journey to Egypt and Babylonia. Trusting Greek tradition, we may assert that many early Greeks, urged by their notorious and childlike curiosity, traveled extensively in the East and profited enormously by their travels. Greek mathe- matics is sufficient evidence of the insatiable hunger of the awakening Greek mind for exact knowledge, and the most adequate measure of its intellectual capacity. Txuo supreme achievements All the immaterial riches of the generous East were anyone’s for the asking and the taking. The early Greeks appear to have asked for everything and to have taken nearly all, fools’ gold along with the rest. In their youthful eagerness to acquire, they overlooked two obvious opportunities, each of the first impor- tance for the futures of science, mathematics, and philosophy then possible. The sixth century before Christ was the time, and Greece the place, for human beings to reject once for all the pernicious num- ber mysticism of the East. Instead, Pythagoras and his fol- lowers eagerly accepted it all as the celestial revelation of a higher mathematical harmony. Adding vast masses of sheer 54 THE DEVELOPMENT OF MATHEMATICS numerological nonsense of their own to an already enormous bulk, they transmitted this ancient superstition to the golden age of Greek thought, which passed it on in the first century a.d. to the decadent arithmologist Nicomachus. He, enriching his already opulent legacy with a wealth of original rubbish, left it to be sifted by the Roman Boethius, the dim mathematical light of the Middle Ages, thereby darkening the mind of Chris- tian Europe with the venerated nonsense, and encouraging the gematria of the Talmudists to flourish like a weed . 4 It is customary in the histories of science and mathematics to ignore these vagaries of the human mind. However, it would seem to impartial onlookers that only a distorted image of the not too flattering facts can result from any historical account which reports only what are now considered successes and ig- nores the failures. Frequently the sense of one epoch has become the nonsense of a later, and what no longer is meaningful may once have been of prime scientific or social importance. A case in point is the phlogiston theory of heat; another, the tripartite infinity of theology, a lineal descendant of the mystical arithme- tic of the Pythagoreans; and yet another, the Platonic theory of mathematical truths, long since abandoned by unmystical mathematicians. Without some attention to such misadventures in ideas, the development of mathematics, no less than that of science, appears as an uninterrupted parade of triumphs with never a recession to relieve the glorious monotony. Gratifying as such a presentation may be, it is not therefore necessarily adequate. Had the Pythagoreans rejected the number mysticism of the East when they had the opportunity, Plato’s notorious number , 5 Aristotle’s rare excursions into number magic, the puerilities of medieval and modern numerology, and other equally futile divagations of pseudo mathematics would probably not have survived to this day to plague speculative scientists and be- wildered philosophers. Nor would a mathematical astronomer 0 of the early twentieth century have beheld the astounding spectacle of God masquerading as a mathematician. Among the gains accruing from the ancient numerology is the inspiration for much of Plato’s theory of Eternal Ideas. If on the other hand the early Greeks had accepted and understood Babylonian algebra, the time-scale of mathematical development might well have been compressed by more than a thousand years. But to a people just starting to grow up mathe- FIRMLY ESTABLISHED 55 matically, the attractions of a mystical, all-embracing philos- ophy were naturally more seductive than those of an austere algebra. A greater disaster at the height of the Greek golden age held mathematics and science back immeasurably. Instead of follow- ing the bold lead of Archimedes and developing a fluent, dynamic mathematics applicable to the ceaseless flux of nature, the lesser Greek mathematicians of the third century b.c., and after, lingered behind with the Piatonists and cast their thought in geometric shapes as perfect and as rigidly static as the Parthe- non. In the entire history of Greek mathematics, all but the incomparable Archimedes and a few of the more heterodox sophists appear to have hated or feared the mathematical infinite. Analysis was thwarted when it might have prospered. Such are the debits in the account of Greek mathematics with time. They are heavy enough; but beside the credits, they are of little moment. It has fallen to the lot of but one people, the ancient Greeks, to endow human thought with two outlooks on the universe neither of which has blurred appreciably in more than two thousand years. From all the mass of their great achievement, these two, each of superlative excellence, may be exhibited here by themselves, not to diminish their magnitude by a crowd of lesser masterpieces, all great but not the greatest. The first was the explicit recognition that proof by deductive reasoning offers a foundation for the structures of number and form. The second was the daring conjecture that nature can be understood by human beings through mathematics, and that mathematics is the language most adequate for idealizing the complexity of nature into apprehensible simplicity. Both are attributed by persistent Greek tradition to Pythag- oras in the sixth century before Christ. No contemporary record of these epochal advances survives; and there is an equally persistent tradition that it was Thales in the sixth century B.c. who first proved a theorem in geometry. But there seems to be no claim that Thales, earliest of the ‘seven wise men of Greece,’ proposed the inerrant tactic of definitions, postulates, deductive proof, theorem as a universal method in mathematics. Again, in attributing any specific advance to Pythagoras himself, it must be remembered that the Pythagorean brotherhood was one of the world’s earliest unpriestly cooperative scientific societies, if not the first, and that its members assigned the common work of 56 THE DEVELOPMENT OF MATHEMATICS all by mutual consent to their master. It is sufficient to remember that these advances were made as early as 400 b.c. at the latest, and that both were Greek. Chronology of Greek mathematics Before considering in some detail a few items of more than antiquarian appeal, we shall give a short prospectus of the lead- ing schools of Greek mathematics with their dates, a few key names, and brief mention of the principal advances made by each. Some of these will not be noted again. All dates except those attached to men’s names are only approximate; those •without a.d. are b.c. The birth, maturity, and senescence of Greek mathematics cover about ten centuries, roughly from 600 b.c. to a.d. 400. The earliest period, 640-550, was that of Thales (624?-550?), of the Ionian school, and Pythagoras (569?-500?). Its outstanding achievements are the founding of mathematics as a deductive system and the program of mathematicizing natural phenomena. In the fifth century, the Greek sophists of Elea in Italy hardly constituted a mathematical school, yet were of funda- mental importance for the development of all mathematical thought. By his ingenious paradoxes on infinite divisibility, Zeno (495 ?— 4-35 ?) cast doubt on some of the reasoning of his predeces- sors, and was partly responsible for the characteristically Greek course which mathematics entered with the succeeding school and followed thereafter. The sophist revolt against plausible reasoning thus marks one of the cardinal turning points in the history of mathematics. The third and fourth schools, Athens and Cyzicus, 420-300, are one except geographically. Of the very first importance for all the future of mathematics was the disposal of some of the sophists’ objections by Eudoxus (408-355), a pupil and at one time a friend of Plato, in his theory of proportion. Essentially a theory of the real number system, this Greek work of the fourth century b.c. was not substantially modified till the latter half of the nineteenth century a.d., when critical difficulties in analy- sis necessitated a thorough reexamination of the concept of real number. In this period, Plato (429-348) was to assume a mathematical importance greatly in excess of any warranted by his own slight contributions. The general professional opinion is that Plato s too rigid ideal of mathematics 7 as a high philosophic art was to FIRMLY ESTABLISHED 57 cramp and trammel mathematicians abler than himself. How- ever, discounting the excessive purity of the Platonic ideal, Menaechmus (375 ?— 325 ?), a pupil of Eudoxus and reputedly a tutor of Alexander the Great, inaugurated the geometry of conic sections. There is a tradition that Plato encouraged Menaech- mus. If this is true, Plato made a fundamental contribution to mathematics. A basic technique was added to mathematical reasoning in this period by Hippocrates s (of Chios, 470 ?-?; not to be confused with the great physician of the same name, of Cos). By exploiting it in his own geometry, Hippocrates demonstrated the power of the indirect method ( reductio ad absurdum , reduction to an absurdity or deduction of a contradiction from an assumed hypothesis which it is desired to disprove). The universal validity of this method remained unchallenged till the twentieth century, when objections were raised to its indiscriminate use in reasoning about infinite classes. The fifth school was the First Alexandrian, 300-30, in the city founded by Alexander the Great in 332. This was the cul- mination of Greek mathematics. With the exception of Dio- phantus (possibly not a Greek, date conjectured from second to fourth century), the rest is anticlimax. In this great age, Euclid (365 ?— 275 ?) wove elementary plane and solid synthetic geometry into the close deductive system that was to remain the school standard for over 2,200 years. He also systematized the Greek arithmetica as it existed in his time, and wrote on geo- metrical optics. In this age lived Archimedes (287-212), the greatest scientific and mathematical intellect of the ancient world, also, by virtue of the uncramped freedom of his methods, the first modern mathematician. Not till England produced Newton in the seven- teenth century, and Germany Gauss in the nineteenth, did the exact sciences show this ancient Greek his peers. With magnifi- cent indifference to the mathematical proprieties of his age, Archimedes used whatever came to mind or hand to advance mathematics. Unlike many of his fellow Greeks, he did not dis- dain experiment. He founded the mathematical sciences of statics and hydrostatics. He anticipated the integral calculus, also, in the one problem of drawing a tangent to his equiangular spiral, the differential calculus. In this age also lived the supreme master of the synthetic method in geometry. Apollonius (260 ?— 200 ?) left but little for 58 THE DEVELOPMENT OF MATHEMATICS his successors in that method to do in the metric geometry of conics. Astronomy became a ' mathematical science during this period, in the work of Hipparchus (of Rhodes, second half of second century b.c.). From Hipparchus on through Ptolemy (second century after Christ), Copernicus (fifteenth century), Tycho Brahe (sixteenth century), and Kepler (sixteenth century), astronomy did not deviate from the Hipparchian program of a geometry to describe the motions of the planets. With Newton, this geometry evolved in the seventeenth century into dynamics. Hipparchus also was the first to use a sort of trigonometry systematically, and is said to have produced the equivalent of a rudimentary table of sines. Geodesy advanced in this period with the measurement, as accurate as the available data and instruments permitted, of a degree of the earth’s surface by Eratosthenes. This man is re- membered also for his reform of the calendar and for his method of sifting out the primes from the sequence of all the integers. Finally, this rich period produced one of history’s most ingenious scientific engineers, Heron (of Alexandria, second cen- tury, perhaps not Greek). The formula [j(j - a) (s - b)(s - c)]* for the area of a triangle with sides a,b,c and 2s s= a -f b + c, often attributed to Heron, is of course important in trigonom- etry. But its peculiar historical significance is elsewhere. It marks a carefree departure from the too rigid niceties of ortho- dox Greek mathematics, a departure which was halted all too soon. No academic Greek geometer would have presumed to ‘multiply together four lines,’ as in the formula; for the product has no geometrical meaning in Euclid’s space of three dimen- sions. The engineer Heron was not deterred by such obstacles. He discovered or transmitted the right result and, like the Egyptian he may have been, left it for future generations of mathemati- cians to show that he had not erred in his own proof. If, however, as is now claimed, the formula is due to Archimedes, the mystery of it all vanishes. The sixth and last school was the Second Alexandrian, 30 b.c.-a.d. 640. The first date marks the absorption of Egypt by Rome, the second 9 the destruction by the Moslems of what little Roman virility, Greek neglect, and early Christian intolerance FIRMLY ESTABLISHED 59 had left — some say nothing — of the great library at Alexandria. But Greek mathematics had lost most of its creative power long before the library disappeared, and only three men in the six centuries of the Second Alexandrian school would have been noticed as mathematicians by the giants of the First. Ancient astronomy culminated in the second century a.d. in the eccentrics and epicycles of Ptolemy. For about fourteen centuries, Ptolemy’s geocentric description of the solar system was to be accepted as ultimate. Geometry and arithmetic had long since become independent provinces of mathematics ■when Ptolemy, compelled by exigencies of astronomical computation, all but split oft trigonometry' as a distinct mathematical science in his geometrical theorems equivalent to the addition formulas for the sine and cosine, and in his computation of a table of chords. The failure of trigonometry to attain its freedom was due to Ptolemy’s lack of algebra and the disabilities of logistic. Almost at the end of the creative period, a belated geometer, Pappus (second half of the third century), either transmitted or himself discovered three prophetic theorems. He proved the focus-directrix property for the ellipse, parabola, and hyperbola, thereby foreshadowing the general equation of the second degree for all conics in analytic geometry. He also proved in effect that the cross ratio (or anharmonic ratio) of four collinear points is a projective invariant, thus isolating a cardinal theorem in the projective geometry of the seventeenth and nineteenth centuries. Finally, he used one of the numerous dis- guises of the integral calculus to obtain the theorem often ascribed to P. Guldin (a.d. 1577-1643, Swiss) that the volume generated by a plane figure F rotated about a fixed axis is AL , A — the area of F, L = the length of the path traced by the centroid of F. Obviously it is impossible to give an acceptable proof without a full use of the calculus. At last, in the rudimentary algebra of Diophantus, also in his higher arithmetic, mathematics all but entered a renaissance. But it was getting late, and the Greek spirit was too tired to return to its point of departure and resume the march begun by others some twenty-four centuries before in Babylon. If there is such a thing as the Zeitgeist, it must have permitted itself a sardonic smile as it prepared the sourest jest in the history of mathematics. Not Diophantus, but his historical predecessor of the first century after Christ, the numerologist Nicomachus, was to transmit arithmetica to Christian Europe. 60 THE DEVELOPMENT OF MATHEMATICS Number from Pythagoras to Diophantus A few items in the foregoing prospectus overtop the rest in importance for what was to be the future of mathematics. These will now be examined in closer detail. The Pythagorean brotherhood’s conception of mathematics was broad and human. All of their philosophy, of which mathe- matics was only a subordinate if important part, was directed to but the one end of sane, civilized living. Arithmetic, geometry, astronomy, and music were the four divisions of their mathe- matics. This tetrad was to survive for centuries, passing through the Middle Ages in the attenuated quadrivium which formed four-sevenths of a liberal education, the rest being the trivium of grammar, rhetoric, and logic. By then, however, the Pytha- gorean liberality of spirit had been stifled, and living often was neither sane nor civilized in any sense that Pythagoras would have recognized. Of Pythagoras himself only legends remain. In middle life he migrated from his native Samos to Crotona in southern Italy, where the best work of his brotherhood was done. For the rest, fable makes him an enthusiastic if somewhat pompous mysta- gogue who had traveled extensively in the East, and who used his mystical lore to impress everybody from blacksmiths to young women. In liberality of mind he was centuries ahead of his time. As but one indication of many that the Pythagoreans sought to enlighten their contemporaries, women were admitted to the master’s lectures; and Pythagoras himself seems to have had no use for the very peculiar masculinity of the Athens of Socrates and Plato. It is said that Pythagoras and his immediate disciples perished in the flames kindled by those whom they had striven to deliver from brute ignorance, prejudice, and bigotry. In any event, the Pythagoreans were driven out to seed their wisdom elsewhere. The evil that accrued from the Pythagorean numerology has been sufficiently noted. But some good also issued from it at a very long last. From nonsensical hypotheses the Pythagoreans deduced that both the sun and the moon shine by light reflected from a Central Fire. Some twenty centuries later, Copernicus (or his officious editor) in his dedicatory epistle to the reigning Pope stated that this wild deduction gave him a hint for his own heliocentric theory of the solar system. Again, readily discovered relations between the positive FIRMLY ESTABLISHED 61 integers of certain categories, such as the odd or even, or the polygonal numbers, might easily delude an imaginative mind into ascribing human and superhuman powers to number. When Pythagoras discovered the ratios -f, ■§, -jj for the lengths of plucked strings under the same tension to give the octave, the fifth, and the fourth of a note, the first recorded fact in mathe- matical physics, it was an understandable extrapolation that “Number rules the Universe,” and that the ‘essence’ of ail things is number. In the pardonable enthusiasm of that too-inclusive generalization, the modern theory of the continuum of real numbers originated. To follow the clue from Pythagoras to the present, we must return to Thales. He too had learned much from the wise men of the East. The story that he predicted a solar eclipse in 585 b.c. appears to be apocryphal. 10 The like may be true for the equally famous and mathematically more important legend 10 that, while in Egypt, Thales estimated the height of the Great Pyramid by an obvious application of similar triangles to the shadow of the pyramid and that cast by his staff when held perpendicular to the ground. Whether or not the legend records a fact, the Pythagoreans by the fifth century b.c. had reached a critical stage in the development of the number concept. For they proceeded to prove that if a,b,c and a',b',c' are corresponding sides of similar triangles, then a/b — a'/b', b/c — b'/c'. (The remarks on history and proof in the Prospectus arc particularly relevant here.) By the fourth century b.c. it was perceived that the Pythago- rean proof concealed the subtle assumption that the numbers measuring the sides a,b,c,a',b\c r are rational, that is, each is expressible as the ratio (quotient) of two integers. But these sides had been assumed to be of any finite lengths whatever. Hence it had been assumed that there is a one-one correspond- ence between the lengths of straight-line segments and the rational numbers. In particular, it had been assumed that the length of the diagonal of a square whose side is a rational number is itself a rational number. If the side is 1 unit, the diagonal is VI units in length. But the Pythagoreans easily proved that VI is not expressible in the form m/n, where m, n are integers. It would be of great interest to know who 11 first proved the irrationality of VI» but probably we never shall. In reply to a question by Socrates, Thcaetetus says, “Thcodorus was writing out for us something about roots, such as the roots of three or 62 THE DEVELOPMENT OF MATHEMATICS five feet, showing that in linear measurement (that is, comparing the sides of the squares) they are incommensurable by the unit; he selected the numbers which are roots, up to seventeen, but he went no farther; and as there are innumerable roots, the notion occurred to us of attempting to include them all under one name or class.” But this does not settle the vexed question of who first proved the irrationality of V2; and anyone who wishes may still believe without danger of contradiction by incon- trovertible evidence that Pythagoras himself did. All we need keep in mind is that the Pythagoreans by the end of the fifth century b.c. knew that V 2 is irrational. With great ingenuity they approximated to by successive solutions of the equa- tions 2a; 2 — y 2 = +1, Two ways of proceeding lay open. The choice was between some lengths corresponding to no number, or and other positive irrationals being numbers. Choosing the second, the geometers of the fourth century b.c. passed one of the epochal milestones in the history of all thought. “The grand continuum” of analysis, the real number system, was already in view. So also were the paradoxes of the infinite. Unless the new numbers, the positive irrationals, could be incorporated with the positive rationals in a unified domain of ‘numbers’ or ‘magnitudes’ so that all should form a self-consistent system under the operations addition, subtraction, multiplication, and division as then under- stood for the rationals, the newly imagined irrationals would be illusory. Further, operations in the enlarged number system must yield the same results for the rational numbers as before the adjunction of irrationals to the rational number system. The demand for internal consistency in the enlarged domain was automatically imposed, for it had already been agreed that mathematics should not defy strict deduction. No further extension was made until the seventeenth century, when the negative numbers were fully incorporated (but with- out mathematical understanding) into the real number system. About 1800 the final step was taken when the imaginaries were adjoined to the completed real n umbe r system, and the domain of complex numbers (a + b \ — 1, a, b real) was created. In both of these later extensions, the underlying method- ology of generalization and internal consistency had not changed since the fourth century b.c. The Greeks appear to have been guided by subconscious mathematical tact. Explicit formulation FIRMLY ESTABLISHED 63 and perhaps clear understanding of the methodology of extend- ing the number system came only in the late nineteenth century. We shall return to this presently. Thus far only half the project of adjoining the irrationals, as the Greeks saw the problem, had been imagined. They thought in spacial imagery and had generalized their conception of geometrical ‘magnitude’ to include both rational and irrational magnitudes. It did not occur to them immediately that the more difficult half of the project remained undone. They had still to prove that their enlarged system of magnitudes was self-con- sistent; and they appear either to have overlooked this necessity entirely at first, or to have considered it obviously satisfied. Looking critically at what appeared obvious — one of the almost infallible ways of making a fundamental addition to mathemat- ics — they discovered that it was not all obvious. On close inspec- tion they perceived difficulties that have not been completely resolved even today. As already emphasized, the manner in which Eudoxus surmounted these difficulties marks a major turning point in the long history of mathematics. It was impossible for the Greeks or anyone else to understand cither geometry or the real number system without some theory of continuity in the mathematical sense. This incidentally neces- sitated a clarification of the limiting processes, such as exhaus- tion, already described in connection with the Egyptian mensuration of the pyramid. Until such processes were strictly validated it was nonsense to speak of the area of a circle, or of the volume of any solid, or of the length of any line, straight or curved, except only when the numerical measures of such areas, volumes, and lengths were rational numbers. As irrational measures arc infinitely more numerous (to the power of the continuum) than the rational, mensuration and the geometrical theory of proportion scarcely existed before Eudoxus. The necessity for drastic revision was strikingly emphasized by Zeno in four ingenious paradoxes or, as some might say, sophistries. A sophistry in mathematics is a logical argument that some dislike but cannot refute. Zeno’s classic four have probably occasioned more inconclusive disputation than any equal amount of disguised mathematics in history. Zeno’s service to mathematics is so outstanding that it would be interesting to know something of the man himself. Very little has survived. By tradition, he was a pugnacious dialectician with a passion for being different from everyone else. In middle 64 THE DEVELOPMENT OF MATHEMATICS age he was “of a noble figure and fair aspect.” His paradoses are evidence enough of an independent mind, and it is told that his uncompromising intellectual honesty finally cost him his life. He had conspired with the political faction which lost, and met his death by torture with heroic fortitude. The first of his para- doses will suffice here. Zeno argued that you cannot get to the end of a racecourse, because you must traverse half of any given distance before you traverse the whole, and half of that again before you can traverse it, and so on, ad infinitum. Hence there are an infinite number of points in any given line, and “you cannot touch an infinite number one by one in a finite time.” Therefore you will never get to the end this side of eternity. In this and another of the same type (Achilles and the tortoise), Zeno argued against the infinite divisibility of space and time. To show his philosophic impartiality, he devised two equally exasperating paradoxes on the other side: if finite spaces and times contain only a finite number of points and instants, we again deduce consequences contradicted by experience. Mathematics had freely used the concept of infinite divisi- bility. Thus Zeno’s paradoxes, in addition to affording grounds “for almost all theories of space and time and infinity which have been constructed from his day to our own,” showed that geometry and mensuration in the fifth century b.c. needed a new foundation. Eudoxus provided this in his theory of proportion, applicable to any real ‘magnitudes.’ Before recalling how Eudoxus met this ancient crisis, we note the nineteenth-century way out of Zeno’s difficulties. By a sim- ple application of infinite series it is easily shown that the runner will reach his goal and that Achilles will pass the tortoise. But — a reservation of the first importance — the logic of con- tinuity supporting the modern theory of convergence descended in the twentieth century to a deeper level than any that had been explored when the nineteenth-century analysts imagined they had disposed of Zeno’s paradoxes. Eudoxus based his theory on his definition of ‘same ratio:’ The ratio P/Q is said to be the same as the ratio X/Y , when, m and n being any (positive) integers whatever, mX is greater than, equal to, or less than nY according as mP is greater than, equal to, or less than, nQ. If the ratios P/Q, X/Y are the same, P, Q , X, Y are called proportionals. The theory is expounded in FIRMLY ESTABLISHED 65 Euclid’s Elements, Book V. Book VI contains the application to similar figures. Some modern critics, particularly among the French, have been unable to appreciate the radical distinction between this Greek theory of the real number system and that now current, due to J. W. R. Dedekind (1831-1916, German). Beginning in the 1870’s, the criticisms continued well into the twentieth century, especially during the nationalistic fervors generated by the first world war. The later critics appeared to be unaware that Dedekind had disposed of their contentions in 1876. The point of historical interest is that no current theory of real numbers is that which sufficed for the fourth century b.c. The Eudoxian theory of proportion indirectly validated the empirical rule of the Egyptians for the volume of a truncated pyramid, and completed the work of the Pythagoreans on similar figures. It also certified the method of exhaustion and, after Dedekind (1872), the use of the integral calculus in the deter- mination of lengths, areas, and volumes. In short, it provided a foundation for the real number system of mathematical analysis. It may be significant that Eudoxus was another of the great Greek mathematicians who is said to have visited the East. At one time a protege and friend of Plato, he left Plato’s Athenian Academy to found his own school at Cyzicus when Plato — so it is said — began to show signs of most unphilosophical envy and jealousy. But this is hardly credible of the man who composed the Lysis and the Symposium. The italicized phrase in the definition of ‘same ratio’ illus- trates the fact that finality is as hard to reach in mathematics as it is in philosophy. For not all schools of mathematical thought in the twentieth century have admitted ‘any integers whatever’ as a legitimate concept in deductive reasoning. The phrase conceals an infinity of trials on all the integers m, n to test the inequalities mX | nY, mP | nQ. Thus a consisent finitist, if there is one, might say that Eudoxus produced a milder para- dox of Zeno’s racecourse; for “you cannot test for an infinite number of pairs of integers in a finite time.” However, the influential mathematical schools ignore such sophistries, and continue to create new mathematics of great interest and indubitable scientific utility. Having passed this outstanding landmark in the develop- 66 THE DEVELOPMENT OF MATHEMATICS ment of mathematical thought, we shall look forward from it for a moment before proceeding to the next stage in the Greek elaboration of number. Its specific importance in relation to geometry, mensuration, and the real number system has been sufficiently indicated; and anyone with the slightest feeling for great mathematics will admit its greatness without any reserva- tion. Contemplating this masterpiece, mathematicians may be pardoned a little pride that it was their guild which fashioned it. But to leave the Greek masterpiece in sculptured isolation as a monument for all time to the perspicacity of the mathematical intellect would be to give a totally false impression of the manner in which mathematics has developed. Its history is not the record of one brilliant victory after another. Rather is it a somewhat sobering chronicle of intelligence fighting desperately against tremendous odds to overcome the all but ineluctable stupidity of the human mind. That such progress as has been made should have been possible at all is the miracle of the ages. With the detailed example of the Greek attack on incommen- surables (irrationals) set out before them with all the elaborate precision of a copybook for children, mathematicians stumbled about for twenty centuries before imitating the Greek method- ology in their struggle to incorporate negative and complex numbers with the positive real numbers in a single self-consistent system. The irrationals first appeared in geometry, the negatives in arithmetic and algebra, and the imaginaries in algebra. Both negatives and imaginaries entered when it was gratuitously assumed that certain rules of operation, known to produce consistent results in special circumstances, would retain their validity in all circumstances of a superficially similar kind. In Greek geometry, the rules in question were those used in proofs concerning similar triangles with rational sides; the tacit assump- tion was that these same rules would give consistent results for all triangles. In algebra, the negatives and imaginaries entered in an analogous way with the solution of equations; and just as the Pythagoreans were reluctant to grant the status of number to the irrationals, so were the earlier algebraists unwilling to admit negatives and imaginaries as legitimate roots of algebraic equations. The Greeks recognized that they were confronted by a funda- mental problem, isolated it, and solved it. Perhaps the decisive step was their bold hypothesis that a ‘magnitude’ (number represented geometrically) need not be rational in order to be a FIRMLY ESTABLISHED 67 ‘magnitude.’ They generalized the concept of magnitude as it first presented itself to experience and intuition. The algebraists up to the seventeenth century failed to recognize that the nega- tives and imaginaries presented a problem at all. They either blindly manipulated such things when the rules for solving equations turned them out, or rejected them without attempting to justify the rejection. A contemporary of Eudoxus would have wanted to know why some equations produced only intelligible roots, others only some intelligible, and others none. In their lack of common mathematical curiosity, the algebraists of Islam and the European Renaissance were contemporaries of the ancient Egyptians. They wondered and were perplexed, of course; but there they stopped, because they lacked the Greek instinct for logical completeness and generality. It was only in the nineteenth and twentieth centuries that these difficulties ivere satisfactorily met, and then it was by a methodology abstractly identical with that of the Greeks. First, it was ascertained what algebraists were subconsciously striving to do. They were attempting, with no mathematical justification, to include all the reals and all the imaginaries in one system closed under the four rational operations (addition, subtraction, multiplication, division) of common algebra. They actually proceeded on the tacit assumption that such closure was a mathematical fact, namely, that it was self-consistent. This is what they wished it to be, in order to certify their empirical calculations. No progress was made until they followed the Greek lead in geometry and stated explicit postulates for the real and complex numbers, thereby defining the ‘numbers’ presented in algebraic experience. Thus the concept of ‘ number’ was extended to cover all sets closed under the four rational operations. Finally, as will be seen, it was proved in the late nineteenth century that the most general set of this kind, in which xy = 0 only if at least one of x, y — 0, is that of all complex numbers a -f b'Sf’-l, a i b real; and that the only such sets are this itself and certain of its subsets, for example, the set of all rational numbers. When wc reflect that it took the Greeks less than two cen- turies to recognize and reach their goal, we may well wonder whether mathematics today is not starting on another of its two-thousand-year quests in search of simplicity. With all its prolific inventiveness, mathematics seems to have lost some of its youthful directness. Nearly always it is the recondite and 68 THE DEVELOPMENT OF MATHEMATICS complicated which is elaborated first; and it is only when some relatively unsophisticated mind attacks a problem that its deep simplicity is revealed. In their further encounters with number, the Greeks found much that underlies some living mathematics, but nothing, per- haps, of such abiding significance as the work of Eudoxus. Like the similar triangles of Thales which were partly responsible for the Eudoxian theory, one origin of some of the most interest- ing Greek higher arithmetic was in Egypt. The Egyptian ‘rope stretchers’ laid out right angles for the orientation of buildings by means of a triangle of sides 3,4,5. A string of length 3+4 + 5 was marked or knotted at the points 3,4. With this and three pegs a right-angled triangle was obtained in an obvious way. Instead of the particular positive integer solution 3,4,5 of a 1 — W + c 2 , they might have used any other, provided they knew any. The general positive integer solution a, b, c was given by Euclid in his Elements (X, 28, Lemmas). This appears to be the first proved complete integer solution of an indeterminate equation. Whether first or not, it is the germ of vast theories in the modern higher arithmetic and, less directly, of the like in algebra. To complete the record, it must be noted that since 1923 it has been customary to deny that the Egyp- tians ever used 3, 4, 5 to lay out right angles. The argument on which this denial is based appears to run as follows. Because the rope stretchers stretched their ropes for purposes other than laying out right angles, therefore they did not lay out right angles by rope stretching. Further, because right angles were laid out by other means, therefore they were not laid out by — etc. It can be asserted only that the history here may be sounder than its supporting logic. The solution in integers, or in rational numbers, of indeter- minate equations belongs to diophantine analysis. The name honors Diophantus, whose treatise of thirteen books, of which only six survive, was the first on the subject. The Latin transla- tion (a.d. 1621) of this suggestive fragment directly inspired Fermat to his creation of the modern higher arithmetic. It also inspired something much less desirable. Diophantus contented himself with special solutions of his problems; the majority of his numerous successors have done likewise, until diophantine analysis today is choked by a jungle of trivialities bearing no resemblance to cultivated mathematics. It is long past time that the standards of Diophantus be forgotten though he himself be FIRMLY ESTABLISHED 69 remembered with becoming reverence. For the opinion on thiB matter of an expert in both the history and the practice of diophantine analysis, those interested may consult L. E. Dick- son’s History of the theory of numbers, vol. 2, 1920. On another account also this work of Diophantus is memora- ble. It was the first Greek mathematics, if indeed it was Greek, to show a genuine talent for algebra. Following the Pythago- reans, Euclid had given geometrical equivalents for simple identities of the second degree, such as a(a -f b) = a~ -f- ab, (a -f b) 2 = a 1 -f- b- -f- 2 ab, and had solved x 1 -{- ax — a-, a positive, geometrically. Diophantus gave essentially algebraic solutions of special linear equations in two and three unknowns, such as x -f- y = 100, x — y — 40. More important, he had begun to use symbols operationally. This long stride forward is all the more remarkable because his algebraic notation, com- pared to that of today or of the seventeenth century when Descartes practically perfected it, was almost as awkward as Greek logistic. That he accomplished what he did with the avail- able technique places him beyond question among the great algebraists. His operational advance was profoundly significant. In algebra a formula, say a -f- b — c, directs us to perform certain operations on given numbers (or, in modern algebra, abstract marks), here an addition and a subtraction on a , b, c , in a pre- scribed order. That is, algebra escapes from verbal instructions to symbolic directions and ceases to be purely rhetorical. Dio- phantus had even invented a species of minus sign, and permitted a negative number to function in an equation on a parity with positive numbers. He also used symbols for the unknowns and for powers. All this was a long step toward symbolic algebra. It seems probable that some of Diophantus’ algebra was of Baby- lonian origin, although the connection has yet to be traced. Unfortunately for the development of algebra and of mathe- matics generally, Diophantus v'as at least four centuries later than Archimedes. To conclude this account of Greek arithmctica, we may return to its origin in geometry and instance the timelessness of great mathematics by an episode in arithmetic and geometry from the twentieth century. The Pythagorean theorem that x- + y- = z~, where x,y,z are the sides of a right triangle, is the basis of metric geometry in Euclidean space. In the spaces defined by Riemann (1854), 70 THE DEVELOPMENT OF MATHEMATICS the quadratic algebraic form x 2 + y 2 in two variables, x, y is replaced by a quadratic differential form in n variables; n = 4 is the case of interest in relativity. The significance of x 2 -f y 2 = z 2 in diophantine analysis has been remarked. There, this equation is also generalized, and it is required to solve the general quadratic equation in n unknowns, with integer coeffi- cients, in integers. This arithmetical problem, together with that of reducing the general equations of the second degree in the analytic geometry of conics and quadrics to canonical form, suggested (nineteenth century) the purely algebraic problem of reducing a quadratic form in n variables to a sum of squares each multiplied by an appropriate coefficient. Incidentally, this problem is important in dynamics. Toward the close of the nineteenth century (a.d. 1882), a notable advance was made in the diophantine problem by Minkowski, then a youth of eighteen. In treating this problem, Minkowski acquired a mastery of the algebraic theory of the reduction of quadratic forms. Becoming interested about the turn of the century in mathematical electromagnetism, he applied his algebraic skill to special differential forms. By the peculiar accident of his interests, he was the ideal candidate to recast the mathematics of Einstein’s special relativity of a.d. 1905 into a shape which still retains its attractiveness. Between the orientation of the Egyptian temples and the welding of space and time into space-time stretch some four or five thousand years of troubled history. In mathematics the two events appear almost contemporaneous. The postulational method Had the Greeks done nothing more than put a foundation under the real number system, they would have been assured of perpetual remembrance in mathematics. But they did a great deal more. Indeed, ‘Greek mathematics’ inevitably suggests synthetic geometry, and it was in the Greeks’ elucidation of spacial form that many see their greatest contribution. The development of geometry from a practically workable empiricism to a strict deductive science was extraordinarily rapid. The earliest proof in geometry is traditionally ascribed to Thales, about 600 b.c. He is said to have proved, as one of some half-dozen theorems, that a circle is bisected by any of its diameters. A century and a half later the Pythagoreans had gone FIRMLY ESTABLISHED 71 about as far in plane geometry as students today in the first half of an American school course. Among other details, they knew the Pythagorean theorem, the properties of parallels, the angle-sum for any triangle and possibly for any convex rec- tilinear polygon, the principal facts about similar figures; and they had adequate geometrical equivalents for addition, sub- traction, multiplication, division, the extraction of square roots, and the Euclidean solution of x z -f- ax = a-. Some of this, naturally, was within the limitations implicitly imposed by their conception of the number system. Scarcely modified, devices of the Pythagorean graphical arithmetic and algebra survive in the techniques of our drafting rooms. In solid geometry, the Pythagoreans knew at least three of the five regular solids, and possibly all. If they did know all, their faith in Number as the ruler of the Cosmos may have suffered a setback; for the first three solids occur naturally in common minerals that would attract the eye of any geometer, while the dodecahedron and the icosahedron, having fivefold axes, do not occur in nature. (Copper antimony sulfide, or tetrahedrite, and zinc blende crystallize in tetrahedra; galena, rock salt, and fluorite in cubes; magnetite in octahedra. None are rarities.) But a proof that precisely five regular solids are possible requires a well-developed theory of Euclidean space. The proof is ascribed to Thcaetetus, about the middle of the fourth century n.c. Euclid in the same century completed the elementary theory of these solids in his Book XIII, as the superb climax of his geometry. To call these solids ‘the Platonic bodies,’ as some of the Greeks themselves did, not only violates history but also insults mathematics. It is true that Plato describes a familiar construction of the five regular solids from the appropriate regular polygons. But it is also true that he used these solids as pulpits from which to preach Pythagorean numerology With the completion of Euclid’s Elements , Greek elementary geometry, exclusive of the conics, attained its rigid perfection. It was wholly synthetic and metric. Its lasting contribution — and Euclid’s — to mathematics was not so much the rich store of *165 propositions which it offered as the epoch-making method- ology of it all. For the first time in history masses of isolated discoveries were unified and correlated by a single guiding principle, that of rigorous deduction from explicitly stated assumptions. Some 72 THE DEVELOPMENT OF MATHEMATICS of the Pythagoreans and Eudoxus before Euclid had executed important details of the grand design, but it remained for Euclid to see it all and see it whole. He is therefore the great perfector, if not the sole creator, of what is today called the postulational method, the central nervous system of living mathematics. It seems strange that Euclid’s method should have had to wait till the nineteenth century for the only kind of appreciation that counts for anything in mathematics, application. Synthetic metric geometry of course continued in the postulational tradi- tion. But this, apparently, was mere inertia; for it was decades after the explosive outburst of projective geometry in the nine- teenth century before that subject received a sound basis. And it was only in the a.d. 1830’s that any serious attempt was made to provide a postulational foundation for elementary algebra. Not until a.d. 1899, in the work of another great geometer, D. Hilbert (1862-1943, German), was the full impact of Euclid’s methodology felt in all mathematics. Concurrently with the pragmatic demonstration of the creative power of the postulational method in arithmetic, geometry, algebra, topology, the theory of point sets, and analysis which distinguished the first four decades of the twen- tieth century, the method became almost popular in theoretical physics in the a.d. 1930’s through the work of P. A. M. Dirac (a.d. 1902-, English). Earlier scientific essays in the method, notably by E. Mach (a.d. 1838-1916, Austrian) in mechanics and A. Einstein (a.d. 1878-) in relativity, had shown that the postulational approach is not only clarifying but creative. Mathematicians and scientists of the conservative persuasion may feel that a science constrained by an explicitly formulated set of assumptions has lost some of its freedom and is almost dead. Experience shows that the only loss is denial of the privilege of making avoidable mistakes in reasoning. As is perhaps but humanly natural, each new encroachment of the postulational method is vigorously resisted by some as an inva- sion of hallowed tradition. Objection to the method is neither more nor less than objection to mathematics. It may be true that the life-sciences, for example, are still too lush a wilderness for the sowing of a few intelligible postulates here and there, but the attempt has begun, as in the work (1937) of J. H. Woodger. If the Pythagorean dream of a mathematicized science is to be realized, all of the sciences must eventually submit to the dis- cipline that geometry accepted from Euclid. FIRMLY ESTABLISHED 73 Flight from intellectual -prudery As Plato (430-349 b.c.) preceded Euclid (365 r— 275 ? b.c.) by about twenty years, it is possible but improbable that the geometer was influenced by the philosopher. It may be regret- table, but it appears to be true, that creative mathematicians pay little attention to philosophers whose mathematical educa- tion has not gone much beyond the elementary vocabulary. Of all changes that mathematical thought has suffered in the past 2,300 years, the profoundest is the twentieth-century conviction, apparently final, that Plato’s conception of mathe- matics was and is fantastic nonsense of no possible value to anyone, philosopher, mathematician, or mere human being. Not all, however, are iconoclasts of Platonic realism. Some, whose mathematical achievements entitle them to an opinion on the matter, have expressed themselves quite forcibly on the enduring validity of realistic mathematics. G. H. Hardy (1877-, English), for example, stated (1940) his belief that “mathe- matical reality lies outside us, that our function is to discover or observe it, and that the theorems which wc prove, and which we describe grandiloquently as our ‘creations,’ are simply the notes of our observations.” It will be recalled that similar beliefs regarding other intangibles caused some rather unpleasant mischief in the Middle Ages and the Renaissance. Plato himself may not be responsible for the more outrageous absurdities concerning mathematics in his dialogues; there is always the half-mythical figure of Pythagoras in the background. But it was the high poetic quality of the dialogues that preserved the ancient nonsense for later generations of mathematicians and philosophers to admire and imitate. This worked great mischief in geometry. In Platonic realism, the straight lines and circles of mundane geometry are unimportant; it is the Eternal Idea of a straight line or of a circle that alone is worthy of philosophic contemplation. Thus in this particular philosophy the useful abstractness of mathematics is vaporized into a nothing of ethereal beauty that has yet to make its first contribution to geometry. To a Platonic geometer it is self-evident that the Archetypal Circle is more rotundly round than any other curve in the Eternal Mind, also that no Idea is straighter than the Ideal Straight Line in the same everlasting locus. Hence it follows that terrestrial geometry should be restricted in all its constructions 74 THE DEVELOPMENT OF MATHEMATICS to a straightedge and a pair of compasses. If, for example, an angle is to be trisected, it must be done with these implements. It follows also that in comparison with the geometry of straight lines and circles, that of ellipses, parabolas, and hyperbolas is slightly disreputable, or at least not ideally immaculate. A geom- etry using any mechanical contrivances other than the sacro- sanct two was severely reprimanded for ‘‘thus turning its back on the ideal objects of pure intelligence.” It is not surprising that Plato disdained applied mathematics. In his philosophy of mathematics, Plato was the finished intel- lectual aristocrat, purer than the purest of pure mathematicians. Fortunately for both pure and applied mathematics, such an excess of purity, not to say prudery, did not appeal to the real aristocrat Archimedes. Archimedes by himself was an epoch in the development of mathematics. There is immortality enough for a dozen in his great discoveries, and these are overshadowed by the methods which he invented or perfected and which, unfortunately, perished with him. Centuries were to pass before science and mathematics overtook him. The legend of his life is familiar from Plutarch’s incidental account. He was a close friend and perhaps a kinsman of Hiero, tyrant of Syracuse, where he was born and where he died. During the siege of Syracuse by Marcellus in the second Punic w r ar, the mechanical armaments of Archimedes delayed and all but defeated the Romans. When the city fell (212 B.c.), the defense- less old mathematician was killed by a Roman soldier. 'Rome'WOn the war, finally destroyed Carthage ( delenda est Carthago!), and marched on to almost unimaginable heights of splendor, but not in science or mathematics. As bluntly practical as the soldier who dispatched Archimedes, the Romans were the first wholehearted- exponents of virile living and bucolic thinking, and the first important people to realize that a modi- cum of brains can be purchased by those who have only money or power. When they neeaed any science or mathematics not already reduced to easy rule of thumb, the Romans enslaved a Greek. But they blundered when they killed Archimedes. He was only seventy-five and still in full possession of his powers. In the five years or more of which the soldier robbed him, his truly practical mind might have taught the Romans something to ward off the fatty degeneration of the intellect which finally rendered them innocuous. All the work of Archimedes is characterized by rigor, imagi- FIRMLY ESTABLISHED 75 nation, and power. He may rightfully be called the second mathematical physicist in history, and one of the greatest. Pythagoras was the first. In this capacity Archimedes is almost unique, in that he used his physics to advance mathematics. The usual procedure, in which he also excelled, is the reverse. A sample of his great work may suffice to suggest the magnitude of the whole. In applying the method of exhaustion to the mensuration (of both surfaces and volumes) of the sphere, cylinder, cone, spheri- cal segments, spheroids, and hyperboloids and paraboloids of revolution, Archimedes proved himself the complete master of mathematical rigor and the perfect artist. Some of this involved (in modern notation) the evaluation of the definite integrals J 0 sin x dx, J’J (ax -f- x-)dx. His problem of cutting a sphere by a plane so that the segments shall be in a given ratio pre- sented him with a cubic equation of the type x z + ab- — bx-, which he may have solved geometrically by the intersection of conics. The catholicity of his interests is shown by his famous ‘cattle problem,’ which demands incidentally the solution in integers x, y of x- — 4,729,494y :: = 1. Finally, in pure mathe- matics, Archimedes anticipated the method of the differential calculus in his construction of a tangent to the spiral (p = ad) known by his name. His most original work perhaps was in his applied mathe- matics. Here, so far as is known, he was a pioneer. Menaechmus and others had successfully applied the method of exhaustion to difficult problems (Archimedes himself mentions Eudoxus and attributes to Democritus the statement of the result for the volume of a pyramid) ; but none had applied mechanics to mathe- matics. Before Archimedes, no scientific mechanics existed. There may have been empirical rules, but such are in a different universe. His discover}' of the law of buoyancy practically created the science of hydrostatics, and his formulation of the theory of the lever did the same for statics. So powerful were his methods that he determined the positions of equilibrium and stability of a floating paraboloid of revolution in various posi- tions. True to the Greek tradition, Archimedes based his mechanics on postulates. His determinations of centroids were about as difficult as those in a course in the calculus today. For example, he found the centroid of a semicircle, a hemisphere, a segment of a sphere, and a right segment of a paraboloid of revolution. It is small wonder that the Moslems held Archimedes 76 THE DEVELOPMENT OF MATHEMATICS in almost superstitious veneration. There was not his like for two thousand years. Archimedes’ sublime disregard of convention is seen in what is his most curious work. It is the problem, which he solved, of finding the area of a parabolic segment. The proof, of course, is rigorous. It amounts to an integration, somewhat disguised as exhaustion in the official proof. It is the unofficial proof which is of greater interest. This came to light in 1906, when a work by Archimedes describing his heuristic method was found in Con- stantinople. To discover what the required area was, Archimedes translated the problem in geometry into an equivalent in mechan- ics. Having solved the latter, he states that the result has not been “ actually proved.” He then proceeds to give a geometrical proof in which, incidentally, he performs the first summation AC of an infinite series in history. The series is ^ 4“ ”, and he uses o the fact that 4“" tends to zero as n tends to infinity. He had n already summed a finite series, ^ as z . *= i An isolated gem may show that Archimedes was as per- spicacious as he was inventive. To the untutored mind it is obvious that by laying off a given segment, no matter how small, a finite number of times, any point on a line may be reached or passed. It was obvious to Archimedes only that this is an as- sumption which should be stated explicitly as one' of the postu- lates of geometry. He did so; and non-Archimedean geometries, in which the postulate is rejected, were constructed in the nineteenth and twentieth centuries. Like Euclid in his explicit statement of the parallel postulate, Archimedes had the true mathematician’s caution in the presence of the obvious. Modern mathematics was born with Archimedes and died with him for all of two thousand years. It came to life again in Descartes and Newton. Through geometry to metaphysics The negative obligation of mathematics to ancient philos- ophy has been indicated. As will appear when we discuss medi- eval Europe, it is possible that the obligation was reversed in that mathematical desert. For the present, it will be of interest to note how Greek mathematics was indirectly responsible for some profoundly interesting work in epistemology since about a.d. 1920. FIRMLY ESTABLISHED 77 The Greek geometers left undecided four elementary prob- lems that were to defy mathematical ingenuity for over two thousand years. None of the four is of mathematical importance today. Historically, no more prolific problems were ever pro- posed, with the possible exception of Zeno’s. Repeated failures to settle the first three disclosed fundamental difficulties unsus- pected by the ancients, and necessitated a sharpening of the number concept. Unsuccessful attempts for about 2,300 years to dispose of the fourth at last suggested a great advance in mathe- matical methodology which now seems trivially obvious, but which eluded some of the keenest minds in history. The problems are as follows. In each of the first three, with due deference to Plato, the desired construction is to be per- formed wholly by means of a finite number of straight lines and circles. Problem 1. To trisect any angle. Problem 2. To construct the side of a cube whose volume shall be twice that of a given cube. Problem 3. To construct a square equal in area to any given circle. Problem 4. To deduce Euclid’s fifth postulate from the others. The fifth postulate is formally equivalent to the following. Through any point P not on the straight line I. there can be drawn, in the plane determined by P and L, precisely one straight line which does not meet L. Problem 2 is equivalent to demanding a geometrical construc- tion, by the means prescribed, for the real root of a: 3 — 2 = 0; Problem 1 is similar. These two were not settled till P. L. Wantzel (a.d. 1814—1848, French) in 1837 obtained necessary and suffi- cient conditions for the solution of an algebraic equation with rational coefficients to be geometrically constructible in the manner specified. Neither of the cubics concerned satisfies the conditions. Thus the problems were proved to be impossible. If the restriction that the only permitted means are a finite number of straight lines and circles be removed, solutions of Problems 1, 2 are readily obtained, for example by conics, as done by the Greeks, or by linkages. The historical importance of these two is the impetus they gave, long after Greece, to the investigation of the arithmetical nature of the roots of algebraic equations with integer coefficients. Such roots arc called alge- braic numbers; a number which is not algebraic is said to be transcendental. The third problem tapped a deeper spring. By Wantzel’s theorem, if Problem 3 is solvable, its algebraic equivalent must 78 THE DEVELOPMENT OF MATHEMATICS be a finite number of equations satisfying his conditions. The problem will be impossible if 7r(= 3.14 . . . ) is transcendental. In a.d. 1882, C. L. F. Lindemann (1852-1939, German) proved that 7T is transcendental. His proof, with its curious dependence on rational arithmetica, would have delighted Pythagoras. Problem 3, as 1, 2, is solvable when modified to permit the use of curves other than circles. The quadratrix (p, 0 polar equation tt p = 2 rd esc 6) invented by Hippias in the fourth century b.c. for the trisection problem suffices. This, however, is of but trivial interest; the significance of circle squaring is its connection with transcendental numbers. Squaring the circle implies an irrationality of a kind radically distinct from that which taught the Pythagoreans that not all numbers are rational; V 2 is algebraic, tt is not. It would seem a reasonable guess to one exploring the number system for the first time that all real numbers are algebraic, or at least that the transcendentals are extremely rare. Cantor proved in a.d. 1872 that the algebraic numbers are the rare exceptions; the transcendentals are infinitely (to the power of the continuum) more numerous. It is an interesting' exercise to trace the implica- tions of the restriction to a finite number of straight lines and circles in the conditions of the three problems. Problem 4 — to prove Euclid’s parallel postulate — will re- appear when we follow geometry through the nineteenth century. It is one of Euclid’s greater achievements to have perceived that this postulate demands explicit statement as an assumption. The new quirk in methodology which finally disposed of the problem in a.d. 1826 may be described here, as it is one of those profoundly simple, powerful devices which are so obvious after they have once been pointed out that they are first imagined only by minds of the highest originality. A problem which has resisted the best efforts of genius for centuries may be impossible, or meaningless, or improperly posed. The quirk is simply to admit that one of these three may be the fact. This admitted, what seems the likeliest of 'che three is developed mathematically. The parallel postulate was circumvented by the third possi- bility: a self-consistent geometry was constructed without it. Problems 1, 2 evaporated when the suspected impossibility was pursued, and a contradiction was deduced from assumed possi- bility. The problem of squaring the circle suffered the same fate, but was much harder to dissipate. A brilliant application of the quirk to a modern problem was Abel’s proof in a.d. 1824 that the general algebraic equation of FIRMLY ESTABLISHED 79 degree higher than the fourth is unsolvable by radicals. He appears to have been the first to state the methodology explicitly as a general procedure. As an item of historical interest, the Persian poet, mathematician, and connoisseur of wine, women, and song, Omar Khayyam, is said to have conjectured in the twelfth century that the algebraic solution of the general cubic is impossible. He was mistaken. We shall recur to this in a later chapter. Beginning about a.d. 1930, the Viennese school of mathe- matical logicians attacked some of the classical problems of philosophy, particularly metaphysics, by this methodology, attempting to show that the problems were either meaningless or improperly posed. Of course history may prove them as mistaken as Omar was. Needless to say, the attack was vigorously resisted, especially by those who refused to master enough elementary symbolism to enable them to read a proof in symbolic logic. Thus four elementary problems of Greek geometry were partly responsible for a subversive movement in philosophy that would have shocked the ancient Greek philosophers as pro- foundly as it shocked some of the moderns. There was at least a prospect in a.d. 1945 that the incipient revolution — if it may be called that with propriety — might necessitate some revision of accepted epistemology. The non-Euclidean geometry of the nineteenth century that issued from Problem 4 abolished Kant’s theory of mathematical ‘truths.’ Plane, solid, and linear loci Toward the end of Greek mathematics, a hesitant step toward unity and generality was taken by Pappus (probably third century) in his MaOrjfxanK&v <ruvccyuty&v fiifiXla. This collection of eight books, of which only the last six and a mere fragment of the second are extant, was a compendium of much of the mathematical knowledge of its time. The missing parts may have dealt with arithmetic; the six known books include proportion, parts of solid geometry, selected higher plane curves, isopcrimetric problems, spherics, centroids, special curves of double curvature and their orthogonal projections, and finally mechanics, which seems to have signified yet more geometry to the ingenious compiler. Those items of the collection that may have been due to Pappus himself have been called brilliant by competent critics; certainly they display a boldness of con- ception and an uncramped freedom of method reminiscent of Archimedes rather than of Euclid. If a kinematic generation of 80 THE DEVELOPMENT OF MATHEMATICS a curve seemed natural to Pappus, he did not hesitate to avail himself of it. Probably much earlier than his day it had been suspected that the three classic problems of Greek geometry are unsolvable by Euclidean methods, although no Greek mathe- matician is known to have stated the impossibility of Euclidean solutions as a working hypothesis. Accepting the suspected fact, Pappus proceeded to a masterly investigation of the higher plane curves which about five centuries of experience had shown to be sufficient for solving the problems. The spiral of Archi- medes, the conchoid of Nicomedes (second century b.c.), the cissoid of Diodes (same century), and the quadratrix of Hippias (fifth century b.c.) were accorded full geometric status. These outlaws of the rigid classical geometry were shown to be as worthy of serious attention as the hackneyed conics. The con- choid had been invented to solve the trisection problem, the quadratrix for the rectification and quadrature of the circle, and the cissoid for the classic Greek problem of inserting two geometric means between two given ‘magnitudes’ represented as straight-line segments. The conchoid and cissoid are alge- braic curves; the quadratrix is transcendental. Nevertheless, all three are thrown into the vaguely inclusive class of ‘linear’ loci. This ill-defined receptacle held all loci other than the plane and solid. Circles and straight lines were ‘plane’ loci; the conics, ‘solid’ loci, doubtless so named on account of their origin as sections of cones (of the second degree). In passing, one of the decisive achievements of Apollonius was his replacement of the three species of cone, used by his predecessors to obtain the various conics, by the right circular cone of which all are sec- tions. If derivation from conical surfaces was the ground for calling conics solid loci, it seems rather peculiar that Pappus should have cast the quadratrix into the nebulous limbo of linear loci. For two of his most striking personal contributions were his definitions of the quadratrix as orthogonal projections of certain skew curves. In one, the curve is the intersection of a cone of revolution and a right cylinder whose base is a spiral of Archimedes. Here was synthetic geometry in the grand man- ner almost of Archimedes himself. Though the classification of loci as plane, solid, and linear may not seem very significant to a modern geometer, neverthe- less it was a conscious attempt to put some system and order into the chaos of imaginable plane curves. Without algebraic symbolism, little either reasonable or useful was possible, and FIRMLY ESTABLISHED 81 almost nothing general. If Apollonius be granted the use of co- ordinates claimed for him by some of his admirers, this in itself but emphasizes the inadequacies of the clumsy substitutes for a genuine symbolism used by the Greek geometers. That they accomplished so much that has retained its interest after twenty centuries or more is a tribute to their genius rather than a recom- mendation of their technique. But lest we overvalue our own acquisitions at the expense of theirs, we may remember that there is as yet no satisfactory classification of the uncountable infinity of transcendental plane curves. If such a classification is not a dead problem, it is a project for the analysis rather than the geometry of the future. A wrong turning? Greek mathematics stands as one of the half dozen or so supreme intellectual achievements of our race. Its best is now well over two thousand years behind us. Looking back on it in the light of the mathematics that has developed since the early decades of the seventeenth century, we shall try to see it dis- passionately in “the cold beams of the history of learning.” Its two greatest achievements — those traditionally attributed to the Pythagoreans — shine out as clearly as ever, and with them, Euclid’s. Apart from Archimedes, two thousand years ahead of his age, what of the rest ? For better or worse, our technical deveopment of science and mathematics differs radically from that of the Greeks. Their mathematics is intelligible to us, and any modern can appreciate it at what they themselves considered its true value. Ours, be- yond the mere rudiments, would appear to them — with the exception of Archimedes — as conclusive evidence of insanity. A straight line to them, for example, meant a finite segment capa- ble of prolongation; to us a straight line is defined once for all from minus infinity to plus infinity. The limited modes of Greek thought are not ours. With the maturing of elementary algebra in the sixteenth and seventeenth centuries of our era, and the introduction of analytic methods in tire seventeenth, mathematics in its return to number drew closer to Babylon, Egypt, and India than it overdid to Greece after the fall of Alexandria. Except for insistence on proof, our preferences in mathematics as in religion are more oriental than Greek. It may be a hard saying, but it appears to be none the less true, that on the long view Greek geometry was in part a tactical 82 THE DEVELOPMENT OF MATHEMATICS blunder. No necessity compelled Thales, Pythagoras, Euclid Apollonius, and all their disciples to develop the synthetic method exclusively. At the beginning of their arduous mathe- matical journey, two possible roads had been plainly indicated to the Greeks by their Eastern predecessors. Either by conscious predilection or ironic mischance they all took the same turning and hewed their way through tremendous obstacles to the end of a blind alley. The synthetic geometry of conics marks the end of the journey. Further significant progress — that which mathematics fol- lows today — became possible only when the harsh way was retraced or temporarily forgotten, and the road which the Greeks had passed by in the sixth century b.c. was entered in the seven- teenth of our era. Returning to the thought of the East, from which Thales and Pythagoras had started, European mathe- matics detoured almost completely round the territory consoli- dated by the Greek geometers. Resuming a march interrupted twenty-three centuries earlier, mathematics during and after the seventeenth century proceeded with incredible speed to the conquest of world after new world beyond the farthest reach of Greek thought. The geometry of Ptolemy’s Almagest appears to us as an all but superhuman effort of mathematical genius. It and the 387 propositions in the conics of Apollonius are the masterpieces of the synthetic method. But the new science inaugurated by Galileo and Newton in the seventeenth century needed more than one or two masters of mathematics every four or five hundred years if it was to exploit its opportunities with reason- able speed. Few mathematicians who have followed the Grecian proofs in Newton’s Principia believe that all the propositions demon- strated could ever have been discovered in one lifetime by the methods of Greek geometry. There are limits even to the mind of a Newton; and we have his own word for it that he used analytic methods — his calculus — for discovery. The rigid syn- thetic proofs were devised partly to reassure himself but prin- cipally so that he might be understood by others. Still, it might be possible to claim the Principia as a monument to synthetic geom- etry. But not the most generous imagination would concede the dynamics of Lagrange, Hamilton, Jacobi, and Lie to a hypo- thetical application of Greek geometry, although these crea- tions of the eighteenth and nineteenth centuries evolved with, apparent inevitability from the dynamics of the Principia. And; FIRMLY ESTABLISHED 83 last, in reference to his method which Newton saw fit to translate into Greek, discovery after all is more important in science than strict deductive proof. Without discovery there is nothing for deduction to attack and reduce to order. Returning for a moment to the sixth century b.c., we may try to imagine -what mathematical history might have been if the Greeks had taken the Babylonian highway. Like other ‘might have beens,’ this one is futile, except possibly as it may indicate which of several roads we ourselves might the more profitably explore. The orientals had a more catholic taste than the Greeks for number. At least some of the orientals were not terrified by mere magnitude; Indian mythology with its millions of deities, its “tan- gled trinities,” and its aeons of aeons is an adumbration of the mathematical infinite. For that matter, theEgyptian trinity exhib- its some of the seeming contradictions of the modern concept of the infinite, with its one-one correspondence between part and v’hole; and the like is evident in Christian theology, the heir, not of dead Greek mythology, but of oriental religions. A minor but significant indication of the ineptitude of the Greek mind for mathematical analysis is the fact that for centuries it remained content with a system of numeration which, compared with the best of the oriental work, was puerile. It is not definitely known that the Early Greeks were ac- quainted with the advances and speculations in number of other peoples; but from internal evidence it is highly probable that they must have heard of them. There are too many oriental inclusions in Greek mathematics to make the miracle of a curiously partial transmission of knowledge credible. For the sake of our hypothesis we shall assume that the Greeks were not entirely ignorant of what their neighbors to the East had done. Had the early Greek mind been sympathetic to the algebra and arithmetic of the Babylonians, it would have found plenty to exercise its logical acumen, and might easily have produced a masterpiece of the deductive reasoning it worshipped logically sounder than Euclid’s greatly overrated Elements. The hypoth- eses of elementary algebra are fewer and simpler than those of synthetic geometry. The algebraic-analytic method in mensura- tion and geometry was well within the capacity of the Greek mathematicians, and they could have developed it with any degree of logical rigor they desired. Had they done so, Apollonius would have been Descartes, and Archimedes Newton. As it was, the very perfection — for its age, and for long after 84 THE DEVELOPMENT OF MATHEMATICS — of Greek geometry retarded progress for centuries. It was admired, as it merited, by the Moslems -who finally restored it in the Middle Ages to a forgetful Europe; and much of the genius that might have gone into expanding their own arithmetic, algebra, and trigonometry was lavished on translation and commentary. If there is any truth in Bergson’s elan vital, or in Hegel’s philosophy of history, Greek geometry was a splendid disaster for both. This, needless to say, is not the traditional conclusion. The superiority of purely synthetic methods over the algebraic and analytic, as being more intuitive, has been urged by numerous distinguished mathematicians, particularly of the British school, since the time of Newton. And we find the same contention being put forward in the seventeenth and nineteenth centuries for the superiority of the synthetic method in projective geometry over the analytic. No working mathematician would deny the utility and suggestiveness of diagrams ; but that is not the point at issue. It has been said that in geometry the synthetic and the analytic methods are like a pair of hands; and this undoubtedly is true, provided the geometry handled is simple enough. But inspection of a treatise on modern physics, or of one on partial correlation in statistics, even when the analysis may be described in the language of geometry, reveals few synthetic proofs, if any. The like is true of the greater part of living geometry itself. And Lagrange, the great master of dynamics after Newton, prided himself that his analytic mechanics contained not a single diagram. One of the most vigorous defenses of the geometrical method of Euclid, Apollonius, and Ptolemy is that of Thomas Young (a.d. 1773-1829, English), the universal genius who is remem- bered for his contributions to medicine, Egyptology, elasticity, and the wave theory of light. His arguments are frequently repeated even today, especially in the intermediate instruction of science students. By a singular historical irony, Young’s defense was first printed in a.d. 1800, the year which marked the end of the middle period of mathematics and the beginning of the recent. It was republished, together with a slashing attack on the analytic mechanics of Lagrange, in a.d. 1855, the year in which Gauss, the inaugurator of the recent period, died. But if Young after all v r as right, his was a voice crying in the wilderness that few appear to have heard. For better or worse, mathematics in the seventeenth century committed itself to analysis, and the Greek methods became of only historical interest. CHAPTER 4 The European Depression It is customary in mathematical history to date the beginning of the sterile period from the onset of the Dark Ages in Christian Europe. But mathematical decadence had begun much earlier, in one of the greatest material civilizations the world has known, in the Roman Empire at the height of its splendor. Mathemati- cally, the Roman mind was crass. Beyond the cumbersome Roman numerals, which can be called a mathematical creation only by undiscriminating charity, the Romans created nothing even faintly resembling mathema- tics. They took what little they needed for war, surveying, and brute-force engineering from the Greeks they had crushed by weight of arms, and were content. When Julius Caesar reformed the calendar in 46 b.c., it was no Roman who proposed leap year with its extra day in February, but the Alexandrian Sosigenes. The Roman contribution to civilization was in law, government, and peace at the sword’s point. The military Pax Romana began collapsing in earnest in a.d. 410, when the invaders penetrated the city of the Caesars, and the last garrison was recalled from Britain to stand with the defenders against an onrushing flood of barbarians. The debacle of Roman grandeur came about sixty years later, and five cen- turies of darkness descended on Christian Europe. Five years after the recall of the Roman garrisons, a riot in the last capital of Greek learning foreshadowed the centuries of confusion, and marked the end of the first great epoch in creative mathematics. One of the last of the Greek mathematicians was a woman, Hypatia. Like her male colleagues at Alexandria, Hypatia was a critic and commentator rather than a creator. Her death sym- £5 86 THE DEVELOPMENT OF MATHEMATICS bolizes the end of pagan science and mathematics, and the beginning of an age of faith. In 415, when Hypatia died, there were good works more urgent than geometry and arithmetica to be done. The hordes from the north were in need of civilizing and conversion to a gentler religion. To the zealous tillers of this all but virgin field it seemed obvious that the decaying remnants of an effete Greek culture must first be cleared out of the way. Had not Greek intellectual- ism and immorality sapped the virility of Rome? Therefore Greek thought must be swept back into the past. As a representa- tive of the older enlightenment, Hypatia was a conspicuous obstacle in the path of the new. Encouraged by their uncom- promising bishop, the willing Christians of Alexandria effectively removed the obstacle by inducing her to enter a church, where they murdered her in a needlessly barbarous manner. 1 Mathematics lived on, just breathing, in Christian Europe. The next significant epoch was inaugurated in the eighth century by the infidel followers of the prophet Mahomet. European mathematics from Boethius to Aquinas Before passing on to the one thing of any suggestiveness for the development of mathematical thought that may have had a root in the sterile centuries, we must propitiate tradition by doing honor to the learned Europeans of that period whose names adorn the classical histories of mathematics. From a long list of historical celebrities we select the following as a fair sam- ple, with their names, their dates, all a.d. henceforth, and the places where they flourished: Boethius (c. 475-524, Rome, Italy); Isodorus (c. 570-636, Seville); the Venerable Bede (c. 673-735, England); Alcuin (735-804, born at York, labored in France); Gerbert (950-1003, Rome); Psellus (1020-1100, Greece, Constantinople); Adelard (early eleventh century, England); Robert of Chester (early twelfth century, England, Spain). This list may be very considerably lengthened without adding any undue burden of mathematics. No census of the leading European mathematicians of the Middle Ages would be complete without the memorable name of Thomas Aquinas (1226-1274, Naples, Paris, Rome, Pisa, Bologna). Although this Newton of scholastic theology is not usually counted among the elite of medieval mathematics, we shall see that he might be. In contemplating the barren record from Bede to Aquinas, it is well to remember that while European civilization rotted, THE EUROPEAN DEPRESSION 87 another culture, the Moslem, 2 was conserving the Greek classics and developing the algebra and arithmetic of India in prepara- tion for the European Renaissance. We are immediately con- cerned only with the contributions of Christian scholars. Of the European background, it is sufficient to recall the persevering struggle of the church to dominate the people and mildly educate a few of them, and the dawning enlightenment that accompanied the crusades of the eleventh to the thirteenth centuries. The crusades no doubt accelerated throughout a wakening Europe the diffusion of knowledge that began with the Moslem conquest of Spain in 711. These influences are reflected in a gradual change, beginning in the twelfth century, in the character of European mathematics. Medievalists disagree on which century, the twelfth or the thirteenth, was of greater significance in the awakening of Europe. The distinction, if any, is unimportant for mathematics. There was one item of any moment, and only one. Latin versions of Greek mathematical classics, made for the most part from translations by the Moslems into Arabic or Persian, became available to European scholars. While remembering with grati- tude the devoted labors of the translators, we need not forget that translation is not creation. The best of the translations added nothing new to mathematics; the worst, by men who might be erudite scholars but who were wretched mathemati- cians, added only misunderstanding. 3 To see how low mathematics sank, and to guess how low it may sink again if the enthusiasts for all things medieval prevail, we resume our sample from Boethius to Aquinas, and note what some of these giants did. In their own semicivilized times, several of the men cited were conspicuous conservers of such civilization as there was. Some helped to found elementary schools, others taught, while the more thoughtful wrote shabby textbooks and zealously cultivated theological numerology. Before the great depression got well under way, Boethius described the consola- tions of philosophy in a homily which was to solace many in dire need of solace in the Middle Ages. Gerbcrt, one of the more enlightened popes — unjustly accused at one time of collaboration with the Devil 1 — donned the tiara in 999 and steered the church safely through that ominous year 1000 whose widely heralded Satanic disasters unaccountably failed to materialize. In defer- ence to scholarship, it must be recorded in passing that one school of medievalists proves conclusively that no disasters were ever prophesied, while an equally positive school proves conclusively 88 THE DEVELOPMENT OF MATHEMATICS that they were. Whatever the facts, Gerbert wrote on division and on computation by the abacus, collected trifles on polygonal numbers, compiled an alleged geometry from Boethius and another still less enlightened source, and is said to have had a part in popularizing the Hindu numerals. He is also reputed to have been a man of vast learning and acute intellect. Some of his letters reveal him as singularly dense in the most elementary arithmetic. If Gerbert’s contributions to mathematics are passed over in silence, it is for the sufficient reason that he made none, despite the fact that no history of mathematics is complete without his illustrious name. The like applies to Bede and Alcuin , 5 justly reckoned among the pedagogical heroes of the Middle Ages. Climaxing this phase, Psellus seems to be remem- bered chiefly because it is doubtful whether he ever did any- thing at all in mathematics. His introductions to Nicomachus and Euclid, perhaps fortunately for his reputation as a mathe- matician, are of uncertain authenticity. But his version of the quadrivium persisted through the fifteenth century. With Adelard and Robert of Chester, we advance to the next stage. An indefatigable traveler and painstaking scholar, Adelard was an intelligent collector and translator of mathematical classics. The path of the bibliophile in the eleventh century was less smooth than it is now, and Adelard frequently risked his skin to secure his coveted manuscripts. He is credited with one of the first European translations of Euclid into Latin and with a translation of Al-Khowarizmi’s astronomical tables. Adelard’s one putative original contribution to mathematics was an utterly trivial problem in elementary geometry. Robert translated Al-Khowarizmi’s algebra. All of Adelard’s predecessors together managed to keep some semblance of life in the rudimentary mathematics of Christian Europe. Beyond that, the best of these worthy men made only clumsy calculations in the simplest arithmetic, or attempted to approach elementary geometry in a spirit that would have dis- graced a Greek schoolboy of fourteen. The mathematical awak- ening of Europe was due to no effort of theirs, and their illustrious names might be dropped from the history of mathematics with- out loss. But tradition, rightly or wrongly, forbids. We therefore continue our descent to the nadir of mathematics, and follow the learned Boethius into the abyss. It was the elementary schoolbooks of Boethius that set the mathematical pace of the Middle Ages in Europe. Returning to THE EUROPEAN DEPRESSION 89 the Pythagorean synthesis, Boethius expounded a denatured quadrivium of arithmetic, music, geometry, and astronomy. Considered apart from their medieval content, the names of these four divisions of the Pythagorean tetrad are impressive. But could Pythagoras have looked behind the names, he might have been somewhat disappointed. The geometry, for example, made a brave show by starting from Euclid. But it did not get very far. Only the enunciations of the propositions in Book I and a few in III, IV were offered to the eager students. At the lowest ebb of mathematical intelligence, the liberally educated graduated from geometry when they had learned by rote the enunciations of the first five propositions in Book I of the Ele- ments. Later, when more of Euclid became available, ambitious would-be clerics were encouraged to memorize the proofs of these propositions. The fifth, appropriately enough, was nicknamed the Asses’ Bridge ( pons asinoruin ) . Few attempted the hazardous passage over the equal angles at the base of an isosceles triangle. In his arithmetic, Boethius followed the Alexandrian Nicomachus. As we have seen, Nicomachus in his turn had followed Pythagoras at his most mystical, producing a shoddy treatise on the elementary properties of numbers that might have been composed by an amiable philosopher with a passion for numerology. In passing, mathematicians rather deprecate such effusions being called “the theory of numbers” in some of the traditional accounts . 7 To confuse astrology with astronomy would be less wide of the mark. However, Boethius reproduced the sieve of Eratosthenes and offered some amusing trifles on figurate numbers. Proof seems to have had no greater attraction for him than for his master Nicomachus. Boethius also is credited by some controversialists with a problematical intro- duction of the Hindu numerals to supplement the abacus and the counting board of trade. The practical outcome of all this was a cumbersome reckoning sufficient for simple transactions involving money, and for keeping the calendar in order so that the date of Easter might not elude annual recapture. To call any of this computation — or of the debased geometry — mathematics is a gross exaggeration. The significance of mathematics as a deductive system had been forgotten. Science having sunk to the level of superstition, the other half of the Pythagorean vision survived only in the fantastic absurdities of sacred and profane numerology. Number indeed ruled the darkened universe of the European Middle Ages. 90 THE DEVELOPMENT OF MATHEMATICS The erudite Boethius however made another contribution to learning that may have had more influence on the development of mathematics than all the editions of his sorry arithmetic and geometry ever had. He made a part of Aristotle’s system of logic available to European scholars in a Latin translation. What follows is only a speculation. But even so, it is less depressing and possibly less futile than the dreary record of the homunculi mathematici whose saintly lives and lack of works constitute the official history of mathematics in Christian Europe of the Middle Ages. Submathematical analysis To the scientific or mathematical mind up to the twentieth century, the logical disputes which absorbed a major part of the mental energy of the Middle Ages had long seemed the acme of futility. But in the two decades following the world war of 1914-18, all things medieval became more popular than at any time since the rise of modern science. For sufficient reasons, many had lost their illusion of progress. Blaming science, with an occasional diatribe against mathematics, disillusioned ideal- ists groped blindly back to the twelfth century, or even the ninth, seeking an authoritative assurance of a security more satisfying than any science. Should followers of these hopeless travelers to the past equip themselves properly, they may restore to the future a treasure-trove for the history of mathematical thought comparable to that which has been recovered from Babylon. They may discover where European mathematics went underground after the death of Hypatia, and what shape it assumed during its long burial. Two things in particular may be sought with some prospect of reward: the struggle of the Greek philosophical concept of the infinite, attributed to Anaximander in the sixth century b.c., to get itself transmuted into the modern mathematical infinite; the closely cognate struggle of mathematical analysis to get itself born. The mathematical mind was not dead in the Middle Ages. It was merely sleeping. In its uneasy rest it imagined something curiously like mathematics; but it was powerless to throw off its dreams and wake. The theological subtleties and the scholastic quibblings which absorbed the intellect of generation after generation of potential geometers and analysts were the troubled dreams of a torpid mathematics. THE EUROPEAN DEPRESSION 91 Possibly the suddenness with which the mathematical mind awoke after its long sleep will seem a less abrupt discontinuity when scholars shall have had the patience to explore medieval thought in the light of modern mathematics. Somewhere and somehow in the Middle Ages mathematics suffered a profound mutation. When it went to sleep, mathematics was Greek; when it awoke, it rapidly developed into something that was not Greek. The Moslems were not responsible for the change. Their mathematics, with its shunning of the infinite, is no closer in spirit to analysis than is that of the Babylonians. The transition from ancient thought to modern appears to have been more difficult in mathematics than it was in science. There seems to be no clear-cut instance of a mathematical mind in the late Middle Ages two or three centuries ahead of its time, as Roger Bacon’s (1214—1294?) was in science. Even Bacon, all but fully awake in science, was still as fast asleep in the mathe- matics which he eulogized as were any of his European opponents of the thirteenth century. His mathematical reasoning 8 is still that of the Aristotelian scholastics whom he believed he was confounding with his logic: “I say therefore that if matter can be the same in two substances, it can be the same in an infinite number . . . Therefore matter is of infinite power. Wherefore also of infinite essence, as will be proved, and therefore it must be God” — which, of course, Bacon is refuting. He seeks to accom- plish his purpose by an argument based on a Euclidean postulate which the sharper scholastics had rejected. They did not assume, as did he, that “the whole exceeds any of its parts” is valid for infinite assemblages. The similarity between Greek art, from sculpture to architec- ture, and Greek mathematics has often been remarked. There is no need to pursue this resemblance here; either it is felt as more than a vague metaphor, or it is dismissed as having no possible meaning. A like comparison between Gothic architecture and modern mathematics had impressed many not primarily inter- ested in mathematics before O. Spengler exploited his Faustian theory of mathematics since the Greeks. Thus, writing in 1905, Henry Adams said of the cathedral at Chartres that “Chartres expressed ... an emotion, the deepest man ever felt — the struggle of his own littleness to grasp the infinite.” 9 Adams also gives a sympathetic parody of the medieval per- version of elementary mathematical reasoning to the uses of scholasticism, in an imaginary but convincing debate between 92 THE DEVELOPMENT OF MATHEMATICS those two formidable champions of submathematical analysis Abelard (1079-1142) and William of Champeaux (1070-1122)! But as a specimen of Gothic mathematics at its best, his quota- tion from Archbishop Hildebert (eleventh century) is more suggestive: “God is over all things, under all things, outside all; inside all; within but not enclosed; without but not extended; above but not raised up; below but not depressed; wholly above, presiding; wholly beneath, sustaining; wholly without, embracing; wholly within, filling.” This goes far beyond any logic of Aristotle; it all but makes its subject the Excluded Middle. Lest, after this, the project of disinterring mathematics from the dialectic of the Middle Ages seem fantastic, two most signifi- cant facts may be recalled to encourage those who would proceed. Georg Cantor (1845-1918, Germany), founder of the modem theory of the mathematical infinite, was a close student of medieval theology. In this connection, it appears that the tradi- tionally religious type of mind is the most strongly attracted by Cantorian mathematics. The other significant fact is the dis- covery 10 by K. Michalski (Polish) in 1936 that William of Occam (1270-1349, English) proposed a three-valued logic, thus antici- pating to a slight extent the work of non-Aristotelian mathe- matical logicians in many-valued logics since 1920. Aristotle’s is a two-valued logic, the Values’ being ‘truth,’ ‘falsity’ assigned to propositions. In Occam’s logic Aristotle’s excluded middle is admitted. To the gentle and scholarly Boethius with his translation of Aristotle belongs a large share of whatever credit there may be for having put mathematics to sleep in medieval Europe. The rest may be awarded to the hordes of tireless logicians who strove for centuries to weld theology and philosophy into a self- consistent whole. When at last Thomas Aquinas (1227-1274) — “the dumb ox of Sicily,” as he was called by jealous and envious rivals, but master of them all — succeeded, interest in the stupen- dous project had already waned. Roused by the waspish Mos- lems, Christian Europe woke, and turned, possibly with a sigh of relief, to some science and mathematics. Glancing ahead, we recall that in the first week of September, 1939, the medieval mind at last came into its own once more in Christian Europe. It would be interesting to know what our regenerated descendants will remember of our science and mathematics in 2039, and what people, if any, are to be the Moslems of the future. CHAPTER 5 Detour through India, Arabia, and Spain A.D. 400-1300 The sudden rise and the almost equally sudden decline of the Moslem culture in the seventh to the twelfth centuries is one of the most dramatic episodes in history. 1 Here we are interested only in seeing what enduring influence the culture of this period had on mathematics; and we must not let the sudden brilliance of Mahometan civilization, contrasted against darkened Europe, dazzle us into seeing more in Moslem 2 mathematics than was actually there. By 622 the followers of Mahomet were well started on their travels. Their swarming under the green banner was the greatest religious revival on record, its only close competitor being the counter-revival of the crusades in the twelfth and thirteenth centuries with their avowed purpose of supplanting the banner by the cross. From the capture of Damascus in 635, the victori- ous Moslems proceeded to the siege of Jerusalem, taking that holy city in 637. Four years later they had subdued Egypt, incidentally putting the final touches to the destruction of the Alexandrian library. This, however, was only a youthful indis- cretion, as the Moslems were shortly to settle down and become the most assiduous patrons of Greek learning in history. Having subdued Egypt, they next (642) took Persia and all its civilized erudition. Seventy years later (711), the conquerors entered Spain, where they furthered civilization for about eight centuries before 95 94 THE DEVELOPMENT OF MATHEMATICS being expelled by the Europeans they had at last stung awake. In addition to sowing the fertile seeds for centuries of war, they had brought the arithmetic and algebra of India and Greece, and Greek geometry, to Europe. Bagdad on the Tigris under the Abbasid caliphs from 750 to 1258 became the capital of culture in the East, Cordova in Spain the intellectual queen of the West. After the Moors’ defeat (1212), Jewish scholars — many of whom had acquired their learning from the tolerant Moslems — vied with Christian teachers in spreading the science and mathe- matics that were to relegate scholasticism to the limbo of forgettable but unforgotten misadventures in intellection. The final act was delayed till 1936, when the degenerated followers of the Prophet returned triumphantly to Spain under a red and gold banner, to harry the descendants of the people who had driven out their ancestors some three and a half centuries before. During the Moslems’ long absence, Spain had con- tributed nothing to mathematics. With the involuntary depar- ture of the Jews in the late fifteenth century, savage intolerance for all free thought, whether of Jew or gentile, succeeded sane liberality, leaving four sterile centuries as its monument to science. Partial emergence of algebra Perhaps the most significant advance of the period was the gradual emergence of algebra as a mathematical discipline in its own right, all but independent of arithmetic and geometry but closely affiliated with both. Trigonometry also became clearly recognizable as a separate division of mathematics; and some see in the trigonometry of the Moslems their greatest and most original work. Part of its originality may be granted. But for reasons that will appear as we proceed, the trigonometry is not comparable in importance for living mathematics with the Hindu-Moslem algebra, with its frustrated struggle toward operational symbolism. Before trigonometry could function vitally in modern mathematics it, like geometry, had to become analytic. There is no hint of such a transformation before the seventeenth century, and actually it was fully accomplished only in the eighteenth. The Moslem trigonometry is still essentially Ptolemy’s, amplified and refined by some algebraic reasoning and an extensive application of Hindu-Moslem arithmetic to the computation of tables. From its very nature as mathematics of the discrete, algebra could not DETOUR THROUGH INDIA, ARABIA, SPAIN 95 become a province of mathematical analysis; and hence it was beyond disturbance by the analytic upheavals of the seventeenth century. Moslem algebra appears to have evolved from the late Greek, as in Diophantus, and the much sharper technique of the Hindus. Estimates of Indian algebra differ widely, but on two points there is substantial agreement. Proof was as distasteful to the Indian temperament as it was congenial to the Greek; the Hindus were as apt in calculation as the Greeks were inept. Only by an ag- gressively sympathetic scrutiny of Hindu algebra can anything resembling proof be detected. Rules were clearly stated, but the statement of rules is not proof. A third feature of early Hindu algebra strikes a modern observer as extremely curious: the first skillful algebraists seemed to find indeterminate (diophantine) equations much easier than the determinate equations of elemen- tary algebra. The reverse is the situation today. A small sample from Hindu algebra will suffice here to indi- cate the quality of what the Moslems inherited, conserved, and partly spoiled. In the sixth century, Aryabhatta summed arith- metical progressions, solved determinate quadratics in one unknown and indeterminate linear equations in two unknowns, and used continued fractions. Shortly after, Hindu algebra experienced what some consider its golden age, with the work of Brahmagupta in the early seventh century, just as the Mos- lems were about to start on their travels. Brahmagupta stated the usual algebraic rules for negatives, obtained one root of quadratics, and, most remarkable, gave the complete integer solution of ax ± by — c, where a, b, c are constant integers. He discussed also the indeterminate equation ax- + 1 = y-. The last is misnamed the Pellian equation; it inspired Lagrange in 1766-9 to some of his greatest work in pure mathematics. It is fundamental in the arithmetical theories of binary quadratic forms and quadratic fields. Its place in the history of mathema- tics will be noted presently. Again it seems strange that algebraists who did not hesitate to attack problems of real difficulty failed to see completely through simple quadratics. As remarked in connection with Eudoxus, the early algebraists were halted by a deficiency in the Greek logical faculty. Without an extended number system, it was impossible for the Hindus to create much that even resem- bled a scientific algebra. Thus Mahavira in the ninth century unhesitatingly discarded as inexistent the imaginaries he en- 96 THE DEVELOPMENT OF MATHEMATICS countered, without attempting to account for their appearance. Three centuries later, Bhaskara recognized that formalism produces two roots for quadratics, but rejected the negatives. Hindu algebra, however, took a hesitant step toward opera- tional symbolism. To what has already been noted regarding symbolism, the following summary of the principal advances of the Hindus toward symbolic algebra may be added. Critics disagree on how far the Hindus got in this direction, but what follows seems to be established fact. Aryabhatta (sixth cen- tury) suggested the use of letters to represent unknowns. Brahmagupta (seventh century) used abbreviations for each of several unknowns occurring in special problems, also for squares and square roots. A negative number was distinguished by a dot; and fractions were written in our way, but without the bar, thus, 3 . 4* A manuscript assigned to the period 700-1100 displays a cross, like our plus sign, written after the number affected, to indicate minus. Bhaskara (twelfth century) imitated Brahmagupta in the notation for fractions, also in the custom of putting one member of an equation under the other, and in a systematic, syncopated script for successive powers. There was no sign for equality. Brahmagupta also effected the reduction of Diophantus’ three types of quadratic equations in one unknown to the standard form now current. Differences of opinion concern the weight to be given these devices. By the most liberal estimate, the Hindus had the gist of algebraic symbolism as an operational technique proceeding according to fixed rules and in standardized patterns: the tech- nique for solving problems of certain types was indicated in the mere writing of the problems. All the elaborate verbal directions for taking the successive steps toward a solution are explained away as insurance against stupidity. For even at its best, Hindu algebra, in spite of its free use of abbreviations, was still largely rhetorical in that operational directions were not fully symbol- ized. The least generous appraisal admits no advance in method- ology beyond Diophantus. The Hindus themselves appear to have left no record supporting the first estimate. Possibly they imagined the meaning of what they did so obvious as to render comment on the methodology superfluous. Introspection in mathematics is a modern neurosis. There is also the vexed question of how much of the Hindus algebra was their own and how much Greek. Until competent DETOUR THROUGH INDIA, ARABIA , SPAIN 97 scholars reach some shadow of agreement, there is little point in others reproducing their divergent conjectures. With the dis- covery of Babylonian algebra, the dispute seems less likely than ever to be settled in a finite time. There are also — a fascinating possibility — the ancient Chinese to be considered. Did they, or did they not, influence the Sumerians ? Possibly the Sumerians taught the Chinese? Perhaps the Indians taught them all: Or did they all teach the Indians? And what part did Syria play? One type of argument supporting a favored conjecture may be noted in passing. If civilization A is assumed to be older than civilization B , and if in B a certain type of problem was discussed at a later date than in A , it follows that B got the problem from A. This is according to the diffusion theory of culture. ■* On the alternative spontaneous theory, no conclusion can be drawn; and even on the diffusion theory it remains to be shown that the data are uncontaminated by intrusions of spontaneity. The scarcity of documentary evidence complicates the problem. Fortunately for our immediate concern, these profound ques- tions need not be settled before the influence of Hindu arithmetic and algebra on Moslem mathematics can be substantiated. The Moslems themselves admit having translated Hindu works. It is therefore reasonable to infer that the Moslems were influ- enced by the Hindus. Assuming this, we note once more the human propensity to take the longest way home. Like the Greeks in their indifference to Babylonian algebra, the Moslems finally turned their backs on the rudimentary hints of an operational symbolism in the Hindu algebra and in their own, and wrote out everything, even the names of numbers, in full. The Moslem retrogression in this respect was as long a backward step as any in the history of mathematics. Absorbed in the intelligent collec- tion and painstaking examination of numerous interesting speci- mens, they missed the main thing completely. Only in 1489, in Germany, with J. W. Widmann’s invention of -r, — , did algebra begin to become more operationally symbolic than it had been for Diophantus and the Hindus. Before leaving Hindu algebra, we note what is usually con- sidered its high tide. We shall inspect this rather closely, first in the haze of a golden sunset, then in the unsentimentalized light of mathematics. The two appearances are strangely dis- similar; it may be left to individual taste which is preferred. Bhaskara, about 1150, “gave a method of deducing new sets of solutions of Cat 5 -4-1 = y- from one set found by trial.” 5 The 98 THE DEVELOPMENT OF MATHEMATICS problem is that of the so-called Pellian equation: to solve Cx 2 + 1 = y 2 in integers x, y, where C is a given nonsquare integer and xy ^ 0. Bhaskara also discussed Cx 2 + B = y 2 , C B being non-square integers. His very elementary devices have excited the liveliest admiration. Contemplating them in the golden haze, we observe 6 that “the first incisive work [on the Pellian equation] is due to Brahmin scholarship,” and note that this equation “has exercised the highest faculties of some of our greatest modern analysts.” We also see that Bhaskara’s attack on the equation “is above all praise; it is certainly the finest thing which was achieved in the theory of numbers before Lagrange.” 7 The first of these three quotations is a verifiable statement of fact. The second seems to imply that Bhaskara’s “incisive work” is qualitatively comparable to that of “some of our greatest modern analysts.” The third makes Bhaskara’s tenta- tive, partial solution a finer thing in the theory of numbers than Euclid’s direct, complete solution of x 2 + y 2 = z 2 . In the unflattering light of mathematics, it appears that Bhaskara could find any number of solutions provided he was lucky enough to guess one. He possessed no means of determining whether a given Pellian equation was solvable. Nor, even when he had derived further solutions from a lucky guess, could he tell whether he had all solutions. His process for generating solutions from an initial one was ingenious. But he ignored the only points of any difficulty or mathematical interest: the existence of a solu- tion and the completeness of those solutions exhibited. In contrast with this supreme achievement of Brahmin scholarship, we observe what happened when one of “our great- est modern analysts” exercised his “highest faculties” on the Pellian equation. Lagrange admitted that he had to stretch him- self to accomplish what he did. In 1766-9 he settled the problem of existence and gave a direct, nontentative method for find- ing all solutions. Bhaskara was an empiricist; Lagrange, a mathematician. The unromantic conclusion is that Bhaskara fell far below the standard set by Euclid, a standard which was not reached again till Lagrange attained it in the eighteenth century. It seems not unjust to draw the same conclusion regarding the rest of Hindu mathematics. But it is generally conceded that the better Hindu algebraists were far ahead of Diophantus in manipulative skill. This, and their frustrated attempt to create DETOUR THROUGH INDIA, ARABIA, SPAIN 99 an operational symbolism, appear to be the chief contribution of the Hindus to the development of mathematics. Moslem algebra seems to have hesitated between the tastes of Greece and India, choosing the latter in its most creative period, only to lapse into an impossible rhetoric as it became classic in the ninth century with the masterpiece of its most famous exponent, Al-Khowarizmi. Indian algebra was translated into Arabic and Persian by the Moslems; and as Arabic was an important language not only in scholarship but also in commerce and war, Greek and Indian algebra, simplified and somewhat systematized by the Moslems, at last penetrated Europe. If the crusades of the twelfth and thirteenth centuries did nothing more, they helped indirectly to spread algebra, trigonometry, the classics of antiquity, and contagious diseases. Of an impressive list of Moslem translators, commentators, and minor contributors, only two need be mentioned here. Each showed some originality and both, particularly the first, pro- foundly influenced early European algebra. To give him almost his full name, Mohammed ibn Musa Al-Khowarizmi (died c. 8S0) of Bagdad and Damascus produced the first treatise (c. 825) in which occurs an equivalent for our ‘algebra’ — al-jebr ttfalmuquabala , meaning ‘restoration and reduction.’ The refer- ence is to what now would be called transposition of negatives to yield equations with all terms positive, and to subsequent reduc- tion by collecting like powers of the unknown. This appears to have been Al-Khowarizmi’s own idea; the work as a whole is a compost of Greek and Hindu results. His principal advance in the positive direction was an application of the Hindu number- names to the numerical solution of equations. Al-Khowarizmi’s signal progress in the negative direction has been noted. 8 Why he returned to a purely rhetorical algebra unenlivened by any trace of symbolism, seems not to be known. A psychiatrist might say it was the death instinct having its way. All but strangled, algebra survived, thereby demonstrating that more than a resolute attempt at suicide is necessary to deprive mathematics of its life. In what is reputed to be an algebraic masterpiece, Al-Karkhi (c. 1010) continued in the rhetorical tradition. If the fact were not well established, it would be difficult to believe that medieval European algebraists had the persistence to find out what the rhetorical algebraists of Islam were attempting to communicate. 100 THE DEVELOPMENT OF MATHEMATICS Whether justly or not, algebra without symbolism is rather disappointing to the average layman who has been assured that “the Arabs invented algebra.” Unfortunately, an expert knowl- edge of Arabic was never one of the more graceful accomplish- ments of a gentleman, nor even of a scholar or a country squire, as was a slight acquaintance with Latin or Greek in the eight- eenth century. Consequently the mathematician or the his- torian of mathematics competent to form a personal estimate of Moslem algebra has always been a rarity; and of the few who have deigned to share their findings with those innocent of Arabic, some have presented the outcome of their researches in the familiar symbolism of algebra as taught to beginners today. In certain respects those sophisticated versions of the original verbiage resemble beggars masquerading in robes of satin. To appreciate the difference between the original and its modern disguise, the curious should prevail upon a profes- sional scholar of Arabic to read them a verbatim translation of an original document in Moslem algebra. In lieu of this, the excerpt presently transcribed from the English translation of Al-Khowarizmi’s Algebra by F. Rosen (1831) may be exhibited. Rosen’s translation reproduces the original Arabic, so that cognoscenti may savor its quality. The passage quoted is from A history of mathematical notations (1928) by the American historian of mathematics, Florian Cajori. Rosen remarks that “numerals are in the text of the work always expressed by words: Hindu-Arabic figures [numerals] are only used in some of the diagrams, and in a few marginal notes.” The excerpt follows. What must be the amount of a square, which, when twenty-one dirhems are added to it, becomes equal to the equivalent of ten roots of that square? Solution: Halve the number of the roots; the moiety is five. Multiply this by itself; the product is twenty-five. Subtract from this the twenty-one which arc connected with the square; the remainder is four. Extract its root; it is two. Subtract this from the moiety of the roots, which is five; the remainder is three. This is the root of the square which you required and the square is nine. Or you may add the root to the moiety of the roots; the sum is seven; this is the root of the square which you sought for, and the square itself is forty-nine. Of course symbolism of itself is not mathematics, and no amount of beautifully appropriate notation can make shoddy or trivial reasoning look like mathematics. Extensive tracts of mathematics contain almost no symbolism, while equally extensive tracts of symbolism contain almost no mathematics. DETOUR THROUGH INDIA, ARABIA , SPAIN 101 However, as in the above specimen, the total avoidance of symbolism is not always a virtue to be imitated, especially by neophytes. For laymen who may have difficulty in recognizing algebra when it is spread before them, Rosen transposes Al- Khowarizmi’s rhetorical exercise into its symbolic equivalent: * 2 + 21 = lOx; X = W + Vim 2 - 213 = 5 + V(2 5 - 21), = 5 + V4 = 5 + 2 - 3, 7. This particular equation recurs many' times in the early history of algebra. Al-Khowarizmi’s treatise is credited with a large share of the mathematical awakening of Christian Europe; and a twelfth- century Latin translation of a lost tract of Al-Khowarizmi’s on the Hindu numerals is said to have done much to acquaint Euro- peans with that great invention. Giving this labor of transmis- sion its full historical weight, and balancing it against Moslem algebra, we may leave the reader to find his own point of equilib- rium somewhere among the three following estimates of Moslem mathematics.” “The greatest mathematician of the time [early ninth cen- tury], and, if one takes all circumstances into account, one of the greatest of all times was Al-Khowarizmi.” In the next two, full weight is given to Moslem trigonometry, to be described presently. “Their [the Moslems’] work was chiefly that of trans- mission, although they developed considerable originality in algebra and showed some genius in their work on trigonometry.” “If the work produced [by the Moslems] be compared with that of Greek or modern European writers it is, as a whole, second- rate both in quantity and quality.” Three centuries after the great Al-Khowarizmi had finished his labors, and therefore toward the close of the cultural period, the Persian poet-mathematician, Omar Khayyam (died c. 1123), reached a considerably higher mathematical level than any of his predecessors. This devil-may-care, somewhat cynical philosopher had imagination. Not content with collections of rules, Omar classified cubic equations and devised a method of geometrical solution for numerical cubics, general within the limitations of the existing number system. Others, said to have taken the hint from Archimedes, had solved cubics by the use of conics long before Omar; indeed, the method was familiar to the Moslems of the ninth century. It was not Omar’s technical labors, how- 102 THE DEVELOPMENT OF MATHEMATICS ever, but his erroneous conjectures that cubics are algebraically, and quartics geometrically, unsolvable, that mark him out as more than a faithful transmitter and a skilled tactician and algebraic taxonomist. But bold and original as he was, Omar steadfastly refused to accept negative roots. His hyperbolas, too, were deficient in negative branches. Once more it was the failure to come to grips with the number concept that thwarted both algebra and geometry. The emergence of trigonometry In its literal meaning of ‘triangle measurement,’ trigonometry is as old as Egypt, of course in an extremely rudimentaryform. Greek astronomy demanded spherical geometry, and this, com- bined with the reduction of observations, necessitated what we should call the computation of trigonometric functions. Ptolemy in the second century after Christ summarized in his (jxeydcXr] crbvrafi'i =) Almagest the main features of spherical trig- onometry, and indicated a method for the approximate calcula- tion of what amounts to a crude table of sines, or ‘half-chords.’ Ptolemy used chords; the crudity was unavoidable by the geo- metrical method necessitating interpolations over too wide an interval. Thus, traditionally, plane trigonometry was merely a computational adjunct to spherical trigonometry, and hence the mathematically more important elements of trigonometry emerged with unnecessary slowness. Perhaps the ultimate source of the Hindu and Moslem development of trigonometry was not applications to surveying but the astronomical necessity for sharper interpolation. A Hindu work of about the fourth century advanced con- siderably beyond Greek trigonometry in both method and accuracy, giving a table of sines calculated for every 3.75° of arc up to 90°. The rule used to compute the table was erroneous, but possibly it gave results of sufficient accuracy for the inexact observations of the age. In any event, its reversion to empiricism affords an interesting illustration of the radical distinction between the Greek mathematical attack and the oriental or, for that matter, between the oriental and the modern even at its most crudely practical. The Moslems adopted and developed the Indian trigonom- etry. Their first notable advance was due to the astronomer DETOUR Til ROUGH INDIA, ARABIA, SPAIN 103 Al-Battani (died 929) in the ninth century. If not actually the first to apply algebra instead of geometry exclusively to trigo- nometry, this astronomer-mathematician was the earliest to take a long stride in that direction. In addition to the Hindu sine, he used also the tangent and cotangent. Tables for the last two were computed in the tenth century, when also the secant and cosecant made their appearance as named trigo- nometric ratios. As the concept of a function was still about six hundred years in the future, none of his work bears much resemblance to elementary trigonometry as it is today. Three more names may be cited as marking definite stages in the emergence of trigonometry as a distinct mathematical dis- cipline. Abul-Wefa in the latter half of the tenth century began the systematization of all the trigonometry known at the time, and reduced it to a decidedly loose deductive system. The first Moslem text on trigonometry as an independent science was that of the Persian astronomer Nasir-Eddin (1201-1274). The book was more than a mere compendium, giving abundant evidence of a sure mathematical talent. Like the algebra of Dio- phantus, this work fell too close to the end of its cultural epoch to exert its full weight on the future of mathematics, and Euro- peans duplicated much of it without, apparently, being aware of its existence. The last name we shall cite suggests a curious bit of history which might be worth exploring. Leonardo of Pisa (Fibonacci) will reappear in the sequel; here we note that he published his masterpiece, the Liber abaci , in 1202 (revised, 1228). Leonardo was largely responsible for acquainting an awakening Europe with the Hindu-Moslem algebra and the Hindu numerals. Among other significant trifles in Leonardo’s book is the well- known algebraic identity (a 2 + b")(c- + d 2 ) = (ac ±bd ) 2 - f (ad + be) 2 . It would be interesting to know where Leonardo picked this up on his travels in the East, as it is easily shown to include the addition theorems for the sine and cosine. It also became a germ of the Gaussian theory of arithmetical quadratic forms, and later of interesting developments in modern algebra. With the appropriate restrictions as to uniformity, continuity, and initial values when a,b,c,d are functions of one variable, the identity contains the whole of trigonometry. 104 THE DEVELOPMENT OF MATHEMATICS Mathematics at the crossroads While Europe slept and all but forgot Greek mathematics, the Moslem scholars were industriously translating all they could recover of the works of the classic Greek mathematicians. Several of these translations became the first sources from which Christian Europe revived the mathematics it had all but let die. For this timely service to civilization, the Moslems no doubt deserve all the gratitude they have received. But even at the risk of appearing ungracious, any mathematician must temper gratitude with the hard fact that scholarship and creation are in different universes. Had the Moslems done nothing but preserve and transmit, they would scarcely have merited a passing mention in even the briefest account of the development of mathematics. This may seem too brutually direct; yet, by the only standard whose main- tenance insures progress rather than stagnation, it is just. The one criterion by which mathematicians are judged is that of creation. Unless a man adds something new to mathematics he is not a mathematician. By this standard, the Moslems were not mathematicians in their extremely useful work of translation and commentary. Remembering that we are interested chiefly in things that have lasted, we shall consider briefly the Moslems’ work of translation and commentary, and with it some of their trigonometry, in the light of living mathematics. Only a few specialized historians of mathematics ever really digest any of the Greek masterpieces. Not only is life too short for those who would acquire some usable mathematics for them to master Apollonius or Archimedes; it would also be as wasteful an effort as could be imagined. Nothing could possibly come out of it but erudition. The works of the Greek masters were washed up centuries ago on the banks of the living stream; the spirit of their essential thought, and some few results that beginners learn today more easily by modern methods, alone survive. Thus the Mosem contribution of translation and commentary has lasted, not mathematically, but only as a monument to scholarship. It can be argued that without this moribund mass of Greek mathe- matics, there would have been no inspiration for the new mathe- matics of the seventeenth century: without Apollonius there would have been no Descartes, without Diophantus no Fermat, and so on, and on. Against this it can be maintained that origi- nality was smothered under a blanket of erudition, and that, DETOUR THROUGH INDIA, ARABIA, SPAIN 105 barring sentimental reservations, the greatest service the dead past can render mathematics is to bury its dead. The arguments are equally incapable of objective decision; so we shall leave them with the fact that mathematicians no longer study the Greek classics preserved by the Moslems, nor have they for the past two and a half centuries. The fate of spherical trigonometry, on which the Moslems lavished so much of their skill, illustrates the inevitable recession of things first developed for immediately practical ends. Their utility may remain; but any living scientific interest they may once have had has long since died. Spherical trigonometry is no longer in the living stream of mathematics, either for its content or for any method it may have. Unless a student today requires the subject for some definite routine, such as the old-fashioned positional astronomy, he need not even know that spherical trigonometry exists. Of all the subjects in elementary mathe- matics, spherical trigonometry is probably the deadest and the most repulsive to anyone with the faintest stirrings of a feeling for vital mathematics. At rare intervals some optimistic enthu- siast attempts to breathe a little life into the dry bones; but after a few perfunctory rattles, silence descends once more, and spherical trigonometry is deader than ever. Even the profound revision (1893) of the entire subject in terms of nineteenth- century algebra and analysis by E. Study (1862-1922, German) attracted only passing attention from mathematicians. Plane trigonometry with the Greeks and the Moslems was encouraged principally because it was a useful servant to its elderly spherical sister, and she in her turn was honored mainly for her sendees to astronomy. While plane trigonometry was growing up, astronomy was cbiefest of the sciences and the only one demanding any considerable application of mathematics. Astronomy then needed only the solution of triangles. When positional astronomy receded to a subordinate routine in modern astronomy, the trigonometric functions — for reasons in no way connected with the solution of triangles — became the indispensa- ble mathematical aid from celestial mechanics to spectroscopy. As modern science evolved after Galileo, astronomy became but one science of many, some being of perhaps even greater practical importance than astronomy in a scientific civilization. Here again it was the trigonometric (or circular) functions that were to prove indispensable, and again for no reason even remotely concerned with the solution of triangles. The sine and 106 THE DEVELOPMENT OF MATHEMATICS cosine derive their scientific importance from two properties: they are the simplest periodic functions; they furnish the first instances of a set of orthogonal functions. Both properties were centuries in the future when the Moslems had finished their work; the second had to wait for the integral calculus. Orthog- onality underlies modern applications of the sine and cosine, making possible, as it does, the solution of important boundary- value problems arising from the differential equations of mathe- matical physics. It would be difficult to imagine a physical science without orthogonal functions; but our successors may, and they may then look back on us as we look back on our predecessors. May they be as mindful of us as we are of the Hindus and the Moslems for what they invented, developed, and passed on to us to be further developed. As we take leave of the Moslems, we see them hesitating rather forlornly at the crossroads of ancient and modern mathe- matics. Progress passed them before they could make up their minds to turn their backs on the past they had rescued from oblivion. CHAPTER 6 Four Centuries of Transition 1202-1603 The thirteenth to the sixteenth centuries in Europe is one of the most eventful periods in world history'. These four centuries also include the sharply marked transition from ancient mathe- matics to modern, die break being clearly discernible in the half-century following 1550. As will appear in the Italian solu- tions of the cubic and quartic equations of about 1545, algebra then was still in the Greek-Hindu-Moslem tradition. The French work (Vieta) of the latter half of the sixteenth century' was in a totally different spirit, and one which mathematicians today' can recognize as akin to their own. In less than fifty' years, the Greek and middle oriental traditions became extinct in creative mathematics. 1 The precise dates 1202, 1603 in the heading are intended merely to recall two of the most significant landmarks in the four centuries of transition. The first marks the publication of Leonardo’s Liber abaci; the second, the death of Vieta, the first mathematician of his age to think occasionally as mathemati- cians habitually think today'. The somewhat narrow scope of Vieta’s technical achieve- ments is irrelevant to his importance in the development of mathematics. It was not what he actually accomplished in mathematics, although that was considerable, that counted; it was the quality' of his thought. Whether or not the mathe- maticians of the early seventeenth century' consciously looked back on Vieta as their herald, he was. They quickly surpassed what he had done; but their superiority' was one of degree, not of kind. 1C7 108 THE DEVELOPMENT OF MATHEMATICS Mathematics was ripe for the transition a full two centuries before ft actually happened. The sharp change was delayed by a social chaos in which civilization had all it could do to keep alive. Concurrently, deeper movements were sweeping the super- ficial barbarism of the times back into the past, and clearing the way for a more humane economy in which mathematics shared. A brief recapitulation of the main events responsible for the delay and the subsequent advance will make the transition seem less miraculous than it otherwise might. Wc shall point out in passing what some of the major events implied for the future of mathematics. Opposing currents All the learning of the ancient world could not continue to flood into Europe for much longer after 1200 without setting up opposing currents. Broadly, the conflict became a struggle be- tween established authority to maintain its vested interests un- impaired, and a quickening impatience with mere authority as the final arbiter between free inquiry and dictated belief, whether in knowledge of the natural universe or in government and reli- gion. With both the rapid assimilation of the ancient learning and "the process of the suns” the thoughts of men were being broadened, and nothing short of complete destruction of the race could halt progress. As it happened, disaster was averted by a rather narrow margin. On the side of liberality, the universities of Paris (1200), Oxford, Cambridge, Padua, and Naples were founded between 1200 and 1225. Although the early universities bore but little physical remcmblancc to what they later became, they were extremely significant steps toward intellectual freedom. The thirteenth century also saw the founding of the great orders of the Franciscans and the Dominicans, at least part of whose activities were educational. A too strict devotion to scholasticism in the early universi- ties precluded any serious study of mathematics; but the phe- nomenon of thousands of eager students at Paris squatting in mildewed straw and avidly absorbing Abelard’s (1079-1142) hair-splitting dialectics and his humanistic contempt for mathe- matics shows at least that the capacity for abstract thought was not extinct. Some of the universities being direct outgrowths oi the cathedral schools, it was but natural that they should favor the curricula they did. As late as the fifteenth century, only a FOUR CENTURIES OF TRANSITION 109 smattering of arithmetic and a few propositions of Euclid satis- fied the mathematical demands of a liberal education as certified by a bachelor’s degree from Oxford. Throughout all this period, war was a capital industry. The sack of Constantinople by the Crusaders in 1204, barbarous though it was in itself, might be reckoned with the cultural gains, as a convincing demonstration that greed and religion form a highly explosive compound. There appears to be no doubt, how- ever, concerning what finally issued from the Holy Inquisition, established in one of its milder forms in 1232, shortly after the Moors were disciplined in Spain. Nor is there great difference of opinion in democratic countries about the protracted dissolution of feudalism, and the faint hints of democracy in the conse- quent rise of the middle and merchant classes in the two cen- turies following 1250. Although civilization almost dissolved in the process, the decay of feudalism and the gradual concretion of national monarchies accelerated the growth of knowledge after the critical period was safely passed. The confusion and intolerance already evident in the thir- teenth century became worse confounded in the fourteenth. But the picture was not painted wholly in one color. The names of Dante (1265-1321) and Petrarch (1304—1374) suggest that an occasional ray of light penetrated the gloom; while that of Boccaccio (1313-1375) recalls that some could still appreciate a bawdy story even in the presence of the Black Death (1347- 1349), which carried off between a third and a half of the popula- tion of Europe. The science which less than three centuries in the future was to be fought with all the weapons of intolerance has alone wiped out such plagues. In this fateful century the Hundred Years’ War also got well under way, lasting from 133S to 1453 by one count, and from 1328 to 1491 by another. Whichever estimate is correct, there seems to be general agreement that war flourished somewhat rankly in Christian Europe for nearly two centuries. For ruthless brutality, cynical disregard of the pledged word, and unblushing degeneracy, the famous Hundred Years of the fourteenth and fifteenth centuries had no superior till the twentieth. Not the will, but only the lack of adequate means of destruction, pre- vented a complete return to barbarism. It seems incredible that anything faintly resembling civilization could survive such a reversion to brutchood. But it did. Had some poet of the blackest years sung “the world’s great age begins anew,” as Shelley did 110 THE DEVELOPMENT OF MATHEMATICS shortly before the deepest squalor of the Industrial Revolution, they would have called him mad. One great scientific invention in the first half of this same fourteenth century of our era passed almost unnoticed except as a curiosity for a few careless years, when Europeans quite suddenly envisioned the limitless horizons of destruction revealed by gunpowder. Radical improvements in the art of war conse- quent on this warmly appreciated gift of alchemy were to necessitate much refined pure mathematics and higher dynamics in the accurate calculation of trajectories. Without the mathe- matics of exterior ballistics, old-fashioned gunpowder, or even modern high explosive shells and rockets, would be less effective than the bows and arrows of the English archers at Agincourt. It therefore seems unlikely that war could be abolished by suppressing mathematics. Although it could not have been foreseen at the time, the fifteenth century was to prove a landmark in mathematics as it was in all knowledge. In 1453 Constantinople fell to the Turks, and Eastern culture found its most hospitable welcome in Italy. The powerful family of the Medici in this period rendered dis- tinguished service to civilization by their patronage of scholars and collectors of manuscripts. So far as mathematics is con- cerned, the net gain of this liberality was a further increase in erudition. But something of infinitely greater importance than the accumulation of libraries happened at this time, and at last made mathematics accessible to anybody with the capacities to take it. About 1450 the printing of books from movable types started in Europe. In the first fifty years of European printing, Italy alone produced about 200 books on mathematics. During the next century the output was slightly over 1,500. The majority of course were elementaiy textbooks; but when a work with some real mathematics in it was printed, it became public property instead of the choice possession of a few who could afford a handmade copy. This was the second of the three major advances in the dissemination of mathematics. The first has already been noted in the transition from oriental secrecy to Greek free thought. The third was delayed for nearly four hundred years after the second, until 1826, when the first of scores of low-priced, high-grade periodicals devoted exclusively to mathematical re- search appeared. Printing also furthered mathematics through its economic insistence on a uniform, simplified symbolism. FOUR CENTURIES OF TRANSITION 111 Toward the close of this century, the discovery of America (1492) implied possibilities for mathematics that nobody could have predicted. The necessity for accurate navigation in mid- ocean, and the determination of position at sea by tables based on dynamical astronomy, indicate the connection between 1492 and Laplace’s celestial mechanics completed only in the first third of the nineteenth century. Some of the fundamental work (Euler’s) of the eighteenth century in the lunar theory was undertaken to meet the need of the British Admiralty for re- liable tables. The stimulus for these particular advances, origi- nating in the voyages of Columbus and others, was about evenly divided among exploration, land grabbing, commerce, and the brutal struggle for naval supremacy. From Laplace’s develop- ment of the Newtonian theory of gravitation in his dynamical astronomy, issued the modern theory of the potential and much of the analysis of the partial differential equations of physics in the nineteenth and twentieth centuries. Thus a modern mathe- matician, whether he lives in the United States or in China, who devotes his life to problems in potential theory with increasingly bizarre boundary conditions owes part of his livelihood indirectly to Columbus. So also does a mathematical physicist who com- putes perturbations in atomic physics; for the theory of pertur- bations was first elaborated in dynamical astronomy. The sixteenth century was equally pregnant with great things for the future of mathematics. The names of Leonardo da Vinci (1452-1519), Michelangelo (1475-1564), and Raphael (1483-1520), three of the foremost among a host, will recall what this critical age, the century of Copernicus (1473-1543), was in art; while those of Torquemada (1420-1498), Luther (1483-1546), Loyola (1491-1556), and Calvin (1509-1564) may suggest what it was in the higher things of life. Cardan (1501 — 1576) published (1545) his Ars magna, the sum and crown of all algebra up to his time, only two years after Copernicus, on his deathbed, received the printer’s proofs of his epoch-breaking Dc revoluttonibus orbium codestium. The impact of Copernicus’ work on all thought and on all social institutions is too familiar to require comment here. Mathematically, the Copernican theory was not a complete re- jection of Ptolemy. The circular orbits of the Greeks re- mained, also thirty-four of Ptolemy’s seventy-nine epicycles, and the sun itself had a small orbit. Although Aristarchus had anticipated the heliocentric theory of the solar system, Coper- 112 THE DEVELOPMENT OF MATHEMATICS ' nicus was profoundly original in his provision of a reasoned basis for what had been only a prophetic conjecture. If any one man is to be remembered as the precursor of modern mathematical- physical science, Copernicus has as good a claim as any. Two further advances stand out as portents of the mathe- matics and science that were so shortly to become recognizably like our own. Stevinus (Simon Stevin, 1548-1620, of Bruges) is usually considered by physicists as the outstanding figure in mechanics between Archimedes and Galileo. Among other things, he stated (1586) the parallelogram of forces in the equiva- lent form of the triangle, and gave a complete theory of statical equilibrium. Modern statics is usually said to have originated with Stevinus. He also had as clear ideas of fluid pressure as was possible without the integral calculus. Incidentally, it may be noted that the development — such as it was — of mechanics by the predecessors of Stevinus is less readily evaluated than is the concurrent progress in mathematics. Reconsideration of medieval contributions to statics has usually deflated the first excessive claims in behalf of some more or less obscure writer to have anticipated Stevinus and even, on occasion, Galileo. Such was the case, for instance, with the suddenly inflated reputation of Jordanus Nemorarius (first half of thirteenth century) as a mechanist of high rank. It is now the opinion of mathematicians and physicists who have had the patience to sift his rhetoric that Jordanus was as unintelligent as his con- temporaries in his conception of mechanics. The other man of science in this period whose work was to influence mathematics indirectly but profoundly two centuries after his death was William Gilbert (1540-1603), physician to Queen Elizabeth of England. Except for some of its attempts at theory, Gilbert’s De magnete (1600) was a thoroughly scien- tific treatise on the behavior of lodestones and other magnets. After the consequences of Newtonian gravitation had been elaborated, A. M. Ampere (1775-1836, French), Gauss, G. Green (1793-1841, English), and others in the first half of the nineteenth century created the mathematical theory of magnet- ism. Either the subject was inherently more difficult than gravi- tation, or less able mathematicians attacked it. Twice as long was required to breach it as had been needed for gravitation. Of course the immediate utility of the Newtonian theory may have enticed the leading mathematicians of the eighteenth cen- tury away from Gilbert’s work; and there is the human possi- FOUR CENTURIES OF TRANSITION lx 3 bility that the dynamics of inaccessible heavenly bodies appeared as a grander project than the attraction of magnets that could be weighed in the hand. Thus Laplace gave the sublimity of celestial mechanics as his chief reason for devoting his life to it, and he was by nature anything but sentimental. However, he did not always mean all he said. All of these men of science of the sixteenth century are over- shadowed in mathematical significance by one not usually reckoned nowadays with the professional mathematicians. Thirty-six years of Galileo’s (156T-1642, Italian) life fell in this period of transition from ancient mathematics to modern. As the universally recognized founder of modern science, Galileo influenced all mathematics, pure and applied. There was thus no lack of scientific daring in the sixteenth century', whatever the traditional custodians of consciences may have thought of it all. It may be said here once for all that some of the custodians did not always look kindly' on young science struggling to free itself from the fetters of authoritative tradition, scholastic as well as ecclesiastic. Human nature being what it is, there is nothing remarkable in any' of this. Both sides believed they' were right; and the side which had all of the power save that of indomitable courage believed that it also had all the right. Conflict was inevitable. It -was savage enough, but no more so than its belated echo in the third and fourth decades of the twentieth century', when science in some of the European states once more found itself fighting for its existence. As for the hostility' to science in the period of transition from ancient to modern thought, it is a mistake to blame one Christian sect rather than another. Anyone who cares to search the record may verify for himself that shades of creed were not the fundamental difference between those ivho welcomed science and those who sought to drive it out. The dissension lay' deeper, in the ageless and irreconcilable antagonism between old minds and y'oung, between those who can accept change and those w'ho cannot. In the sixteenth and seventeenth centuries, the y'ounger mind finally won its freedom and retained it for over two hun- dred years. During those brief centuries of free thought, science and mathematics prospered, and life for the majority on that account was less indecent than it was in the days of the Black Death and the Hundred Years’ War. It would be astonishing if mathematics had failed to respond to the crosscurrents of so tempestuous a transition from the old 114 THE DEVELOPMENT OF MATHEMATICS to the new as the four centuries from 1200 to 1600. But for the greater disasters for which human stupidity was only partly culpable, the response might have come two centuries earlier than it did. A terminus in algebra All through this period, as in the preceding, geometry con- tinued to stagnate. Beyond translations of the Greek classics, such as the Latin editions of Euclid (1482) and Apollonius (1537), the work in geometry did not rise above the level of what would now be exercises in elementary textbooks. The Greek methods appeared to be exhausted, and mathematical progress was wholly in the divisions of arithmetic, algebra, and trig- onometry. At the beginning of the period, arithmetic and algebra were still confused in a loosely coordinated alliance; at the end, they were satisfactorily divorced. Trigonometry also gained its liberty from astronomy in this period. The Liber abaci (1202) of Leonardo of Pisa (c. 1175-c. 1250) has already been mentioned. This famous book by a man who was not by training a scholar at last converted Europe to the Hindu arithmetic. Leonardo himself is better known in mathe- matics by his other name, Fibonacci (son of Bonaccio). The son of a warehouse official, Fibonacci traveled for business and amusement in Europe and the Near East, observing and analyz- ing the arithmetical systems used in commerce. The obvious superiority of the Hindu numerals and the Hindu-Moslem methods of computation inspired Fibonacci’s book; and in spite of outraged protests from conservative mer- chants and the then-equivalents of chambers of commerce, the abacus and the counting board were finally (about 1280) rele- gated to the attic in European trade. Thus Fibonacci is indirectly responsible for the deluge of practical manuals on elementary computation and the flood of commercial arithmetics which have poured from the printing presses of the world ever since the fifteenth century. In spite of their great practical utility, none of these indispensable works has contributed anything of importance to the development of mathematics. 2 Fibonacci also expounded the Eastern algebra with genuine understanding, but otherwise made no advance. His Practica geometriae (1220) gave an equally enlightened treatment of elementary geometry. His original work lay in the borderland between arithmetic and algebra. In his Liber quadratorum FOUR CENTURIES OF TRANSITION 115 / (c. 1225), Fibonacci discussed some special diophantine systems of the second degree, such as x- + 5 = y 2 , x- — 5 = z-, which arc harder than they look. Judged by the standard set by Euclid in his integer solution of x 2 + y 2 = z~, Fibonacci’s work is on a far lower level. It does not seem to have occurred to him that the real problem in diophantine analysis is to find all solutions, not merely some. This failure to sense the generality of a problem is characteristic of the distinction between ancient and modern algebra, also between mathematics and empiricism. Euclid was the one exception in about two thousand dreary years of a de- based theory of numbers; even Diophantus, as we have seen, was content with special cases. Although it is distinctly a minor issue, we must mention Fibonacci’s famous recurring series defined by “b ^n, w 0, 1, . . . , Ho ~ 0, — 1, which gives the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... . Fibonacci encountered this sequence in a problem (which the reader may recover for himself) concerning the progeny of rabbits. There is an extensive literature, some of it bordering on the eccentric, concerning these numbers and their simplest generalization, tt„ + » = au n+ i + bu n , a , b constant integers, the most interesting modern work being that inaugurated by E. A. Lucas (1842-1891, French) in 1878. Some professorial and dilettant esthetes have applied Fibonacci’s numbers to the mathematical dissection of masterpieces in painting and sculpture with results not always agreeable, although sometimes ludicrous, to creative artists. Others have discovered these protean numbers in religion, phyllotaxis, and the convolutions of sea shells. It would be interesting to know who first imagined anything transcendental in Fibonacci’s numbers. Their simplest origin is in the Greek problem of dividing a line in extreme and mean ratio, the so-called golden section. It is said that some of the measurements of Greek vases, also the proportions of temples, exemplify the golden section; and one prominent psychologist even claimed to have proved that the pleasure experienced on viewing a masterpiece alleged to be constructed according to the golden section is a necessary consequence of the solid geometry of the rods and cones in the eve. Fibonacci’s quality as a mathematician emerges unmistak- ably in two isolated items, which also hint at the delay in the development of mathematics consequent on the social chaos 116 THE DEVELOPMENT OF MATHEMATICS that followed him. The first is his use of a single letter, in one instance, to denote a number in his algebra. Possibly this is the earliest definite trace of the generality of algebra as distinguished from mere syncopation and verbally expressed rules for numeri- cal computation. The second item marks Fibonacci as a true mathematician far ahead of his time. Being unable to give the algebraic solu- tion of x 3 + 2x 2 + 10a; = 20, Fibonacci attempted to prove that a geometrical construction of a root by straightedge and compass alone is impossible. He could not have succeeded with what was known at his time. He then proceeded to find a numeri- cal approximation to a root. There was nothing in algebra like the inspiration for the attempted proof of impossibility till the nineteenth century. All through the period of transition, algebra was concerned principally with the solution of equations. After quadratics had been solved within the limitations of the existing number system, the central problem was to find similar, that is, ‘radical,’ so- lutions for the cubic and quartic equations in one unknown. There are two distinct problems in the solution of algebraic equations: to construct, by means of only a finite number of rational operations and root-extractions, performed on the literal coefficients of a given equation, all functions of those coefficients which shall reduce the equation to an identity; to construct a numerical approximation to a root of an equation with numerical coefficients. The first problem is called the solu- tion by radicals, and is the one of greater interest in the develop- ment of algebra. The problem of approximating to a root is the one of importance in applications, for two reasons. As will appear considerably later, solution by radicals is impossible for the general equations whose degree exceeds four; the explicit radical solutions of the cubic and quartic are all but useless in numerical work. The problem of approximation is said to have been effectively solved by the Chinese in the thirteenth and fourteenth centuries a.d. This work, if authentic, excelled most of what Europeans accomplished in numerical solutions 3 until almost the same pro- cedure was reinvented by W. G. Horner (1773-1827, English) in 1819. Unfortunately, like nearly all oriental mathematics except Indian arithmetic and algebra, the Chinese method might as well never have been invented for any influence it had on the develop- ment of mathematics. Neither in the Orient nor in Europe did it FOUR CENTURIES OF TRANSITION 117 start a forward movement, and it cannot be said to have passed into the living stream. Its human interest is the evidence it affords (if authentic) that mathematical talent is not the exclu- sive possession of any one race or of any one people. The solution of the cubic and quartic by radicals was essen- tially completed by 1545. (The implied reservation refers to the lack of understanding at the time of negative and imaginary roots.) The history of this final triumph is spiced by a violence and chicanery that seem a trifle excessive even for the unin- hibited sixteenth century. Cardan (1501-1576), whose name ornaments the solution of the cubic in every intermediate text- book on algebra, obtained the solution from Tartaglia under promise of secrecy and published it as his own in the Ats viagna (1545). This ingenious algebraist is also renowned as an astrol- oger, his masterpiece in that direction being a horoscope of Christ. Among others of his excesses which offend our squeamish modern taste, Cardan is said to have disciplined a wayward son by cutting off his ears. Tartaglia (Nicolo, 1500-1557, Italian; the nickname Tar- taglia means ‘the stutterer’) had intended — so it is said — to crown a projected work of his own with his solution of the cubic. His name commemorates a split palate, which the traditional account credits to a saber slash inflicted by an inefficient soldier who, in doing his duty, was assisting in the massacre of the inhabitants of Tartaglia’s native Brescia. They had taken refuge in the local cathedral. Tartaglia, a boy of twelve at the time, was left for dead but, owing to the devotion of his mother and of the dogs who licked his wounds, recovered. As a mature man, Tartaglia contributed to the obsolescence of sabers by pioneering work in exterior ballistics, investigating (1537) the range of a projectile and stating that the range is greatest when the angle of projection is 45°. The solution of the quartic also harmonized vrith its social background. Its hero, Ferrari (1522-1565), was less fortunate than Tartaglia in his family affections. He is said to have been poisoned by his only sister. But Italy in the Renaissance with- out arsenic would be like veal without salt. In recounting these traditional embellishments of Italian algebra in the sixteenth century, we have tried merely to show that mathematics can live and flourish in what purists might call barbarism, and to suggest that it may survive beyond 2000. 1 here has been no intention of misjudging the protagonists in 118 THE DEVELOPMENT OF MATHEMATICS the drama of algebra by a misapplication of our own domestic ethics. As we live according to our lights, they live according to theirs; and if their domestic relations now seem strangely foreign to us, we have but to glance at modern international relations to feel thoroughly at home. On the domestic front, theft of scientific work is still practiced, unfortunately for struggling young men. 4 The scientific history of the cubic and quartic is involved and possibly not yet thoroughly unraveled. It appears to be as follows. Scipio del Ferro (1465-1526) of the University of Bologna solved x 3 + ax = b in 1515, and communicated the solution to his pupil, Antonio Fior, about 1535. Tartaglia then solved x 3 + px- = q and rediscovered Fior’s solution. Not believing that Tartaglia could solve cubics, Fior challenged him to a public contest, in which Tartaglia solved all of Fior’s equations while Fior failed to solve any of Tartaglia’s. Cardan was not a lame copyist. He was the first to exhibit three (real) roots for any cubic. He advanced beyond the mere formal solution in recognizing the irreducible case (all roots real), when the radicals appearing are cube roots of (general) complex numbers. The first to recognize the reality of the roots in the irreducible case was R. Bombclli in 1572. Cardan also suspected that a cubic has three roots, although he was baffled by nega- tives and imaginarics. His most important advance, however, was the removal of the term of the second degree. This had an element of scientific generality which, apparently, he failed to appreciate fully. Ferrari’s solution of the quartic (c. 1540) also appeared in the Ars magna. The solution is substantially the same as that in textbooks on algebra, leading to a cubic resolvent. These solutions of the cubic and quartic mark a definite end 5 in the algebraic tradition of Diophantus and the Hindus. They were sheer tours de force of ingenuity. Modern mathematics deprecates mere ingenuity and seeks underlying general princi- ples. Proceeding from a minimum of assumptions, a modern mathematician exhibits the solutions of particular problems as instances of a general theory unified with respect to some concept or universally applicable method. An isolated solution obtained by ingenious artifices is more likely to be evidence of incomplete understanding than a testimonial to perspicacity. The ingenious solver of special problems may still be a useful member of mathematical society, in that he turns up mysterious phenomena FOUR CENTURIES OF TRANSITION 119 for abler men to strip of their mystery, but he is no longer regarded as a mathematician; and to call a man who imagines he is a mathematician a problem solver is to offer him the unfor- givable insult. This is one line of cleavage between ancient and modern mathematics. It may be asked why these solutions of the cubic and quartic arc still included in the second school course in algebra. Nobody with any sense attempts to use them for numerical computation, and as encouragements to skillful trickery they are positively detrimental. Instead of these relics of the sixteenth century it would be more to the purpose to offer the unified treatment of quadratics, cubics, and quarries which has been available since 1770-71, when Lagrange set himself the problem of seeing why the ingenuities of Tartaglia, Ferrari, and their successors worked. Such a treatment is considerably easier than that certified by historical tradition. A beginning in algebra and trigonometry Passing by an enormous mass of minor contributions, such as short tables of binomial coefficients, anticipations of Pascal’s arithmetical triangle, improvements in algebraic notation, and the like, we come to the first notable progress toward generality in the methods of algebra and trigonometry, and hence in all mathematics. The slow accretion of details which ultimately passed into elementary mathematics in permanent if modified form is not comparable in importance for the whole of mathe- matics with the striving for uniformity in methods. Success in this direction created a science from what had been little better than a museum of tricks that worked occasionally, although nobody understood or seemed to care why. The transition from the special to the general is first un- mistakably discernible in the work of Vieta (Francois Vietc, 1540-1603, French), who, like Fibonacci, was not a mathe- matician by training or profession. Vieta’s activities ranged from cryptography in the military service to politics. At various times he was a member of parliament in Bretagne, a master of requests at Paris, and a king’s privy councillor. His mathematics was his recreation. Some have stigmatized his work as prolix and obscured by a private jargon. But even if this were true, such superficial defects could not conceal the essential qualities of generality and uniformity in much of what Vieta invented. There is no need to describe Victa’s contributions in detail. 120 THE DEVELOPMENT OF MATHEMATICS His attack on quadratic, cubic, and quartic equations brings out the essentials. In each case he removed the second term of the equation by a linear transformation on the unknown. Cardan had already done this for the cubic; Vieta appreciated the importance of this step as a general procedure. For the cubic he made a second rational transformation, and finally a simple one involv- ing a cubic irrationality. This produced a quadratic resolvent; and hence the solution of the cubic was reduced to the essential steps which the nineteenth-century theory of equations was to prove unavoidable by any device. In this work we see also a germ of the theory of linear trans- formations, whose ramifications were to branch out all through later algebra and thence, in the concept of invariance, through mathematics as a whole. Here also we see a clear recognition of the art of reducing an unsolved problem to the successive solu- tion of problems already solved, and the beginnings of tactical uniformity and generality. Vieta’ s solution of the quartic was similarly scientific, and led to the familiar cubic resolvent. In Vieta’s work we observe again the curious retardation of the number system. Negative roots appear to have been unin- telligible to him, although he noted the simplest of the relations between the coefficients of a given equation and the symmetric functions of its roots. He also considered the possibility of resolv- ing the polynomial /(*) in an algebraic equation f(x ) = 0 into linear factors. Anything approaching completeness or proof in this direction was far beyond the algebra of the time, and in fact was not attained till Gauss in 1799 settled the matter by giving a proof which would be admitted 6 today for the funda- mental theorem of algebra. Letters had been used before Vieta to denote numbers, but he introduced the practice (c. 1590) for both given and unknown numbers as a general procedure. He thus fully recognized that algebra is on a higher level of abstraction than arithmetic. This advance in generality was one of the most important steps ever taken in mathematics. The complete divorce of algebra and arithmetic was consummated only in the nineteenth century, when the postulational method freed the symbols of algebra from any necessary arithmetical connotation. Improving on the devices of his European predecessors, Vieta gave a uniform method for the numerical solution of algebraic equations. Its nature is sufficiently recalled here by noting that it was essentially the same as Newton’s (1669) given in text- FOUR CENTURIES OF TRANSITION 121 books. Although Vieta’s method has been displaced by others, its historical significance is of more than antiquarian interest. The method applies to transcendental equations as readily as to algebraic when combined with expansions to a few terms by Taylor’s or Maclaurin’s series. An algebraic equation of degree 45 which Vieta attacked in reply to a challenge indicates the quality of his work in trigonom- etry. Consistently seeking the generality underlying particulars, Vieta had found how to express sin nd (n a positive integer) as a polynomial in sin 0, cos 0. He saw at once that the formidable equation of his rival had been manufactured from an equivalent of dividing the circumference of the unit circle into 45 equal parts. But for his lack of negative numbers, Vieta would have found all 45 roots instead of the 23 he did. More important than this spectacular feat rvas Vieta’s suggestion that cubics can be solved trigonometrically. Indicative of the general haziness of algebra in the time of Vieta, his partial failure underlines the fact that even toward the close of the sixteenth century there was no clear conception of what is meant by the roots of an algebraic equation. Once more the obscurity arose from an incomplete understanding of the number system of algebra. As evidence of Vieta’s modern or Archimedean freedom, his application of both algebra and trigonometry to geometrical problems may be instanced, particularly in the use of algebra to replace geometric constructions wherever feasible. That he found nothing new in geometry of any lasting importance is immate- rial; it was the boldness of his thought that mattered. But, systematic and general within its necessary limitations as this algebraized geometry was, it would be giving it infinitely more than its due to call it a precursor of analytic geometry in any but the strictly chronological sense. It nowhere even hints at the spirit of analytic geometry which made analysis and geometry complementary aspects of one mathematical discipline. Victa’s principal advance in trigonometry was his systematic application of algebra. In both plane and spherical trigonometry he worked freely with all six of the usual functions, and in the former obtained many of the fundamental identities algebraically. With Vieta, elementary (non-analytic) trigonometry was practically completed except on the computational side. All computation was greatly simplified early in the seventeenth century by the invention (1614) of logarithms. In p re-logarithmic 122 THE DEVELOPMENT OF MATHEMATICS computation Vieta extended (1579) the tables (1551) by G. J. Rhaeticus (1514-1576, German), giving the values to seven places of all six functions for every second of arc, instead of for every ten seconds as in Rhaeticus. The deliberate separation of trigonometry from astronomy is usually credited to Regiomon- tanus (Johannes Muller, 1436-1476, German) in his systematic De triangulis of 1464. Elementary algebra at the close of the sixteenth century had still to receive many perfections, especially in notation, before it became the simple routine of our textbooks. But by 1600 the straight path for all future development had been clearly indi- cated. The way which Vieta had pointed out was followed with brilliant success by a host of workers, most of whom made incidental improvements in algebraic notation and technique while developing their major interests in the new mathematics. There was not another first-rank algebraist after Vieta till the eighteenth century, 5 when Lagrange sought and found deeper levels. To conclude this account of Vieta, we quote the opinion of a first-rate mathematician and mathematical historian on Vieta’s place in the history of mathematics. Writing in 1843, De Morgan expressed himself as follows. Vieta is a name to which it matters little that we have not dwelt on several points which would have made a character for a less person, such as his com- pletion of the cases of solution of right-angled spherical triangles, his expres- sions for the approximate quadrature of the circle, his arithmetical extensions of the same approximations, and so on. The two great pedestals on which his fame rests are his improvements in the form of algebra, which he first made a purely symbolical science, and showed to be capable of wide and easy applica- tion in ordinary hands; his application of his new algebra to the extension of trigonometry, in which he first discovered the important relations of multiple angles; and his extension of the antient rules for division and extraction of the square and cube roots to the exegetic process for the solution of all equations. . . . If a Persian or an Hindu, instructed in the modern European algebra were to ask, “Who, of all individual men, made the step which most distinctly marks the separation of the science which you now return to us from that which we delivered to you by the hands of Mohammed Ben Musa [Al-Kho- warizmi]?” the answer must be — Vieta. . . . When will the writer who asserts that Cardan was substantially in possession of Vieta’s algebra attempt to substantiate his assertion by putting so much as half a page of the former side by side with one of the latter ? The development of symbolism The importance of an easily manipulated symbolism, as implied by De Morgan, is that it enables those who are not FOUR CENTURIES OF TRANSITION 123 great mathematicians in their generation to do without effort mathematics which would have baffled the greatest of their predecessors. The formulas in an engineers’ handbook, for instance, if transposed into concise verbal equivalents, with a liberal use of abbreviations and conventional signs for the most frequently occurring words, might be intelligible to an Ar- chimedes; to the average engineer they would probably be exasperating gibberish. And the prospect of having to combine several such verbalized formulas, in the hope of gaining useful information, might discourage even a modern Archimedes. In mathematics itself, as distinguished from its applications, the situation is the same. Unless elementary algebra had become “a purely symbolical science” by the end of the sixteenth century, it seems unlikely that analytic geometry, the differential and integral calculus, the theory of probability, the theory of numbers, and dynamics could have taken root and flourished as they did in the seventeenth century. As modern mathe- matics stems from these creations of Descartes, Newton and Leibniz, Pascal, Fermat, and Galileo, it may not be too much to claim that the perfection of algebraic symbolism was a major contributor to the unprecedented speed with which mathe- matics developed after the publication of Descartes’ geometry in 1637. It is therefore of interest in following the evolution of mathematics to review the principal stages by which elementary algebraic symbolism reached its present maturity and to note how the lack of an effective symbolism hampered the progress of mathematics in some of its more productive periods. Two general observations may help to clarify the somewhat confused historical record. In his analysis of Greek algebra (1842), G. H. F. Ncssclmann (German) noted three historical phases of algebra, to which he gave the suggestive names rhetorical, syncopated, and symbolic. In the earliest phase, the rhetorical, the entire state- ment and solution of an algebraic problem were wholly verbal. It is not exactly clear why the outcome should be called algebra at all, unless it be that similar problems and their solutions reappeared in a later phase scantily clothed in at least the sug- gestion of a symbolism. The middle phase, the syncopated, was distinguished from the first only by the substitution of abbrevia- tions for the more frequently occurring concepts and operations. Syncopated ‘algebra’ was thus an early instance of the quarter- truth that “mathematics is a shorthand.” If algebra were 124 THE DEVELOPMENT OF MATHEMATICS nothing more than a shorthand, its contribution to the rudi- ments of mathematical thought would not be very impressive. The third phase, the symbolic, presents algebra as fully sym- bolized with respect to both its operations and its concepts. It also does much more than this. Symbolic algebra replaces verbalized algebraic processes, which cost the practitioners of rhetorical and syncopated algebra much patient thought, by symbolic procedures summarizing chains of verbal reasoning in readily apprehended rules requiring only passive attention. The experience gained through centuries of laborious trial is condensed in mechanical processes which can be applied and manipulated with a minimum of thinking. If such manual dexterity as almost suffices for competence in solving linear equations, say, is condemned — as it frequently is — for its all but negligible educational value, it has had the merit of liberating the higher faculties of mathematicians to attack problems more difficult than any that taxed the devious ingenuity of the Greeks, the Indians, the Moslems, and the alge- braists of the early Renaissance. In even elementary mathe- matics there is still opportunity enough for invigorating and profitable mental exercise. Finally, symbolic reasoning, as in the current phase of algebra, has suggested extensive generaliza- tions and economical unifications. A typical example was the introduction (1655) of negative and fractional rational ex- ponents, culminating about two centuries later in arbitrary complex exponents with a satisfactory theory to justify their use. The second general remark regarding the evolution of mathe- matical symbolism is implicit in the recognition of the three phases of algebra. As algebra progressed, a multitude of in- dividual names for members of what came to be recognized as one inclusive class were abandoned in favor of a uniform terminology significant for all members of the class. Further, in several instances, uniformity was possible only because the several members of the class were unified by some underlying property, usually simple when at last uncovered, of the rational numbers. When such was the case, an appropriate numerical character was imposed on the whole class, and an algoristic symbolism, amenable to the operations of rational arithmetic, brought the algebraically important characteristic of the class within the grasp of all but involuntary manipulative skill. For example, when it was finally perceived, after centuries of FOUR CENTURIES OF TRANSITION 125 overlooking the elusive fact which now seems obvious, that the powers x, x 2 , a* 3 , A' 4 , a* 6 , a 6 , . . . arc unified with respect to their exponents 1, 2, 3, 4, 5, 6, , and that multiplication of powers of the unknown is effected by addition of exponents, an incredible mass of confusing terminology and inefficient rules was swept into the past, and with it, an equal or greater mass of tortuous thinking. Similar syntheses, again originating in some concealed but gradually perceived property of the rational numbers, accompanied the growth of the number concept. To cite a simple instance, the equations ax 2 + bx — c, ax 2 = bx + c, with a, b, c positive rational numbers, presented two distinct problems to algebraists before negative rational numbers were handled correctly and with (unjustified) con- fidence. The use of negatives reduced the solution of the two equations to that of the single equation ax 2 -f- bx -f* c — 0, with a, b, c rational numbers. In passing, it is strange to find the two special quadratics treated independently in textbooks of less than a century ago. But perhaps this is not remarkable when we remember that Gauss consistently wrote a -2 as xx, for the curiously unmathe- matical reason that neither is more wasteful of space than the other. For ourselves, we still call x 2 and a * 3 the square and the cube of a*, possibly for easy diction, or perhaps because some of the ancients were mathematically fluent in the limpid jargon of areas and volumes. But there arc already hints that squares and cubes may obsolcscc from the vocabulary of algebra before many more centuries have passed. In the meantime, no tyro in algebra need be seriously discommoded by his inability to sec immediately what squares and cubes have to do with a, provided only we do not afflict him with the corresponding names for a' 4 , a -5 , a 5 , . . . from the geometry of hypcrspacc, as we should do if wc arc to foster linguistic purity. But, as was discovered by too hasty penitents in the Middle Ages, even purity may sometimes cost too much for comfort. The absence of symbolism in Babylonian algebra, already noted, poses the problem of what is to be recognized as algebra in the rhetorical and syncopated phases. Since it seems to be agreed among historians of mathematics that the Babylonians, the Egyptians, Diophantus, and the more rhetorical Moslems actually practiced a more or less rudimentary algebra, it is clear that the absence or presence of symbolism is not the his- torical criterion. More than a mere matter of words is involved. 126 THE DEVELOPMENT OF MATHEMATICS From the frequently decisive impacts of symbolism on the gen- eral development of mathematical reasoning, it would seem that mathematics itself and not the pedantries of terminology is the important issue. An older conception of algebra than that now universal identified algebra with the solution of equations. If this antique be admitted, both the rhetorical and the synco- pated phases are accepted as algebra without further qualifica- tion, and the entire historical development of the subject down to the beginning of the nineteenth century acquires a deceptive unity and a specious coherence. A somewhat similar conception of algebra indicates the use of unknowns, whether verbalized or symbolized, as the historical clue to be followed. This, however, comprises too much, as it includes such geometrical problems as the construction of a circle to satisfy prescribed conditions. To narrow the scope of the ‘unknown’ sufficiently, it may be restricted to the domain of numbers — a restriction which lost its stringency with the invention of analytic geometry. In an obvious sense all mathematics is a quest for the unknown. In addition to affording a clue to the development of symbolism, this inclusive definition has the signal advantage over its too numerous competitors of permitting the algebraists or the anal- ysts to claim all mathematics as their province, as already done by some of the geometers. In spite of all objections, the available data seem to show that for following the historical development of symbolism, either equations or unknowns offer a convenient directive. Again, the mere existence of equations all through the protracted evolution also foreshadowed a most important aspect of mathe- matical thought, which dominated much of the work of the recent period beginning in 1801 , that of mathematics as a study of relations. In the earliest stages, the only relation considered was equality, and it required about three thousand years for this ubiquitous concept to reach full symbolic representation. This may serve as a typical example of the slowness with which the commonest paraphernalia of current mathematics evolved. Operations seem to have been symbolized more readily than relations. If this is a correct statement of the facts, it accords with the order of increasing abstractness. But success in one department apparently did not appreciably stimulate inventive- ness in another, and until the recent period the development of symbolism proceeded haphazardly. In modern mathematics the creation of an efficient notation FOUR CENTURIES OF TRANSITION 12 7 may sometimes have been accidental, but usually it ivas the outcome of conscious effort. An example of the first is the nota- tion j (not a/b) for fractions, an invention whose full value may not have been appreciated by its author. Possibly the most striking instance of the second is Leibniz’ -y- (not dy/dx r) for the derivative of y with respect to x. Nobody in the history of mathematics was more sensitive than Leibniz to the potentiali- ties of a rationally devised symbolism, and nobody gave the ‘philosophy’ of mathematical notation more painstaking thought than he. More recently, the notations a/b for fractions and dy/dx for derivatives in some respects have illustrated progress in reverse. Centuries of easy habit were discarded to accommodate the incompetence of printers — the modern reinventor of a/b gave substantially this reason for his departure from custom. In our own century the incompetence of machines contributed in a similar manner to the delinquence of happily conceived notations, for example that of the tensor calculus, until printers learned that it pays to hire competent engineers to revise their machinery. This is one of the few instances where the economic motive has reacted simply and directly with the ideal of mathematical clarity to the benefit of both. In the matter of -y the motivation was partly a humane desire to mod- erate the repulsivcncss of vulgar fractions to young children. A small sample of the many notations for powers of the unknown will suffice to suggest the progression from rhetorical to symbolic algebra. Ahmcs (seventeenth century b.c.) used a word, variously translated as ‘heap,’ ‘amount,’ ‘mass,’ to denote the unknown. Diophantus (third century a.d.) used a shorthand for the successive powers of the unknown: x z was the ‘power,’ x 3 the ‘cube,’ and ‘power,’ ‘cube’ were denoted by (what were probably) abbreviations of the corresponding Greek words. Say these abbreviations were P, C; then PP, PC, CC, denoted the fourth, fifth, sixth powers of the unknown, and so on. Here evidently there is a rationale behind the syncopation. Such a notation was but ill adapted to the simultaneous repre- sentation of several unknowns. Traces of operational symbolism also arc attributed to Diophantus. Addition was indicated by juxtaposition, subtraction by a special symbol whose genesis is 128 THE DEVELOPMENT OF MATHEMATICS still in dispute — it may have been the first letter of a Greek word, or a genuine operational symbol in the sense that it was not derived from an abbreviation. If the latter, it was a sig- nificant step toward symbolic algebra. But Diophantus did not rise to relational symbolism, using the first two letters of the Greek word for ‘equality’ to denote ‘is equal to.’ It is doubted by some scholars whether Diophantus made any use of symbolism, his claim to having done so resting on a manuscript of his arithmetic written about a thousand years after his death. Both the Indians and the Moslems followed Diophantus in what may be described as additive juxtaposition to denote successive powers of the unknown. Aryabhatta (fifth to sixth centuries a . d .) abbreviated the unknown to ya, its second, third, fourth, sixth powers to va, glia, va va, va glia , and so on. He also provided for several unknowns; ya (the first unknown), ka, ni, pi (the second, third, fourth unknowns) the abbreviations being those of the color names black, blue, yellow. Operations were indicated after the operands by the words ghata, bha, indi- cating addition, multiplication. Thus xy, is ya ka bha, where x, y are the unknowns. The substitute for a sign of equality was adequate, one of two equals being written under the other. The word for ‘root’ was mula. Combined with other words, as varga mula, ghana mula for ‘square root,’ ‘cube root,’ mula was hardly so much a mathematical symbol as a common noun. A closer approach to symbolism was the indication of the negative of a number by writing a dot or a small circle above the number. Of the Moslems, Al-Khowarazmi (first half of the ninth century), used jidir (root) for the unknown, and vial (power) for its square. Al-Karkhi (early eleventh century), with kab for the cube, composed the fourth, fifth, sixth, seventh, . . . powers by juxtaposition, as mal mal, mal kab, kab kab, mal mal kab. The Moslems generally followed Diophantus in simplifying equations by combining like terms. Both appear to have been led astray by this natural simplification. The significant classification of equations in one unknown is not according to number of terms, but by degree. However, the later Moslems, in spite of their ineffectual protosymbolism, recognized that the next problem after the quadratic in equations was the cubic, by no means an easy recognition from their point of view. Late in the fifteenth century, the Moslems approached a purely symbolic representation of an operation in writing only the first letter of the Arabic for ‘root’ above a number to indicate FOUR CENTURIES OF TRANSITION 129 square root. This might be considered an intermediate stage between syncopation and a matured symbolism in which opera- tions are designated by specially devised signs whose verbal origin, if any, is no longer recognizable. An example of the last is the current sign for equality. Unless it were definitely known that Recorde invented this sign ( Whetstone of wiite, 1557), it might well be mistaken for a degenerated form of a word in medieval shorthand. But Recorde denoted equality by = be- cause, it seemed to him, no two things could be “moare equalle” than “a paire of paralleles” — which is reminiscent of the remark that “William and John,” twins, “are very much alike, especially William.” But Recorde had the true symbolist’s instinct for the ultimate perfection, by whatever conceit he chose to propitiate his syncopating contemporaries. The Egyptians had used the hieratic form of their hieroglyph for ‘equality’; the Greeks, the first two letters of their word, the Moslems, the last letter of their word, till they reverted to total verbalism and wrote out ‘equality’ in full. It remained for Recorde to do the right thing. In equations, the passage through the three phases was similar to the evolution of the progressively more algoristic notation for powers, which reached a climax in Wallis’ (1655) .V", A a/n for l/.v n , yjx. Greek equations were partly rhetorical, partly syncopated, with little that would now be recognized as algebraic symbolism. Al-Khowarizmi’s equations were purely rhetorical; a Latin translation of one is “census et quinque radices cquantur viginti quatuor or “the square of the unknown ( census ) and five unknowns ( radices ) are equal to twenty four,” that is x 5 -f- 5x — 24. The Europeans of the sixteenth and seventeenth centuries gradually approached full symbolism in the writing of equations, as seen in the following specimens. Cardan (1545) wrote + 6x = 20 as “ cubus p 6 rebus aequalis 20,” in which there is nothing to indicate that cubus and rebus are powers (third, first) of the same unknown. The p is ‘plus.’ Vieta, with C, Q, N for ‘cube,’ ‘square,’ ‘number’ or ‘unknown,’ also left it to be inferred that these terms refer to one unknown in IC — 8£? + 16iV aequ. 40. At last Descartes (1637) settled the matter (except that he missed x- for xx) by writing .v, xx, x s , x \ **, ... for Vieta’s A 7 , Q, C, QQ, QC , . . . , putting all positive integral powers on the uniform notational basis familiar today. It all seemed so simple when it was finally done after centuries of effort. 130 THE DEVELOPMENT OF MATHEMATICS Volumes might be (and have been) written about the evolu- tion of mathematical symbolism. Probably almost anyone leafing through these will agree that lack of appropriate sym- bolism constrained the Greek arithmeticians and algebraists to consider special cases of what might have been their problems, and prevented the Indians and the Moslems from producing an elementary algebra within the capacities of ordinary adolescents. CHAPTER 7 The Beginning of Modern Mathematics 1637-1687 Historical sketches may sometimes decoy us into artificial divisions of human progress by centuries or half-centuries de- marked by precise dates. Having just passed through one such critical fifty years, we may well suspect that another is more imaginary than real. Be this as it may, the half-century from 1637 to 1687 is universally recognized as the fountainhead of modern mathematics. The first date marks the publication of Descartes’ Geometric, the second, that of Newton’s Principia. 1 From this prolific period, as from the Greek golden age, we shall select only those contributions which overtop a multitude of interesting details in their significance for the development of all mathematics. Some of the items omitted will be noted in later chapters, where they may be naturally included without interrupting the continuity of the main current. Thus, in this period infinite series advanced notably, but were of minor impor- tance compared with the calculus. Again, the contribution of Leibniz to symbolic logic can be best described in the light of modern work, and will be noted in the final chapter. It may be remarked once for all that mathematics overshadows its creators; that we arc primarily interested in mathematics; and that each of the men cited did far more than the few items described here, but that much of what is omitted has for long been of only antiquarian interest. 131 132 THE DEVELOPMENT OF MATHEMATICS As some of the men whose work is to be reviewed were more directly responsible than others for the creation of modern mathematics, they will be given more extended notices than might be justified on purely impersonal grounds. Like Pythag- oras, they too will doubtless vanish as personalities and live only in the body of mathematics as the centuries slip away; but at present they are close enough to us to be more than names attached to mathematical abstractions. These outstanding originators of modern mathematics were not merely half a dozen eminent men in a crowd; they towered above the majority of those who preceded or came after them. Conspicuous eminence in mathematics was harder to achieve after these men had lived, simply because by the power of their methods they had quite suddenly raised the whole level of attain- able mathematics. Geometers, for instance, were no longer condemned to crawl among the five conics and a handful of simple higher plane curves after Descartes had given them wings. It is arguable that even the most original of these men was indebted to the very humblest of those who preceded him. But their incomparable superiority in generality of outlook almost inclines us to regard all of them rather as sudden mutations touched off into explosive activity by accidents of their environ- ment, than as orderly end products of a creeping evolution. Five major advances Modern mathematics originated in five major advances of the seventeenth century: the analytic geometry of Fermat (1629) and Descartes (1637); the differential and integral cal- culus of Newton (1666, 1684) and Leibniz (1673, 1675); the combinatorial analysis (1654), particularly the mathematical theory of probability, of Fermat and Pascal; the higher arith- metic (c. 1630-65) of Fermat; the dynamics of Galileo (1591, 1612) and Newton (1666, 1684), and the universal gravitation (1666, 1684-7) of Newton. With these five, two further departures in new directions may be cited for their influence on subsequent advances: the syn- thetic projective geometry (1636-9) of Desargues and Pascal; the beginning of symbolic logic (1665-90) by Leibniz. Throughout the first half of the century, reactionary hos- tility to science continued its losing fight, reaching its futile climax in the condemnation of Galileo by the Inquisition in 1633, only four years before Descartes, safe in Holland, per- THE BEGINNING OF MODERN MATHEMATICS 133 mitted his masterpiece to be printed. Intolerance mas partly offset by the scientific societies founded during this period or shortly after. Only the three most influential need be mentioned. The Royal Society of London was incorporated in 1662; Newton was its president from 1703 to 1727. The French Academy of Sciences (Academie des Sciences, Paris) crystallized in 1666 from the informal meetings of a group of savants, some of whom, including Mersenne, Descartes, and Mydorge, were primarily mathematicians. The Berlin Academy (Societat der Wissen- schaften) was founded in 1700 at the instigation of Leibniz. He was its first president. The Paris and Berlin Academies have consistently been more cordial than the Royal Society to pure mathematics. The importance of these and other academies for the ad- vancement of science during the seventeenth and eighteenth centuries cannot be overestimated. Together, they did far more than the universities for science, one of their chief functions being the publication of research by their members. Even more important than this was the living example each of the scientific societies afforded of a nucleus of intelligent, influential men in a society still cowed by religious bigotry and scholarly intolerance. By the end of the seventeenth century, science had grown too sturdy for indiscriminate attack; and the forces of reaction, fighting among themselves, lacked the wit to combine against their common enemy. A remarkable feature of the rapid development in mathe- matics was that the continuous and the discrete divisions ad- vanced simultaneously. The advance in the continuous might have been expected; the other has the appearance of an accident. Neither more or less trivial arithmetic of permutations and combinations, nor unsystematized observations on games of chance, offer a sufficient explanation of the sudden and complete emergence of the fundamental principles of the theory of probability. The most prolific of all the new acquisitions was the calculus; for when geometry became analytic, it derived most of its life from the analysis of functions continuous except at isolated singularities. There was thus provided an infinite store of curves and surfaces on which geometers might draw, and to which they applied the methods of the calculus to discover and investigate exceptional points, such as cusps and inflections, not intuitively evident from the equations. 134 THE DEVELOPMENT OF MATHEMATICS It is not surprising that for more than a century after it became public property, the calculus and its applications to geometry, dynamical astronomy, and mechanics attracted all but a few of the ablest men, to the comparative neglect of combinatorial analysis, the theory of numbers, algebra (except improvements in notation and Descartes’ work in equations), symbolic logic, and projective geometry. For more than twenty centuries, geometry and astronomy had dominated mathe- matical tradition in the works of the masters. Now here at last was the universal solvent for all the intractabilities of classical geometry and astronomy, and the philosopher’s stone that changed everything it touched to gold. Difficulties that would have baffled Archimedes were easily overcome by men not worthy to strew the sand in which he traced his diagrams. Leibniz did not exaggerate when (1691) he boasted that “My new calculus [and Newton’s] . . . offers truths by a kind of analysis, and without any effort of the imagination — which often succeeds only by accident — ; and it gives us all the advan- tages over Archimedes that Vieta and Descartes have given us over Apollonius.” The calculus of Newton and Leibniz at last provided the long-sought method for investigating continuity in all of its manifestations, whether in the sciences or in pure mathematics. All continuous change, as in dynamics or in the flow of heat and electricity, is at present attackable mathematically only by the calculus and its modern developments. The most important equations of mechanics, astronomy, and the physical sciences are differential and integral equations, both outgrowths of the seventeenth-century calculus. In pure mathematics, the calculus at one sweep revealed unimagined continents to be explored and reduced to order, as in the creation of new functions to satisfy differential equations with or without prescribed initial condi- tions. One of the simplest of all such equations, dy = f{x)dx, in a sense defines the integral calculus; and the corresponding integral, jf(x)dx, alone suggests an endless variety of functions according to the form of f(x). In the discrete division, continuity is of only secondary importance. Primarily, combinatorial analysis is concerned with the relations between subclasses of a given class of discrete objects, for example with the interrelations of the permutations and combinations of the members of a given countable class. Fermat’s and Pascal’s work of 1654 on probability lifted com- THE BEGINNING OF MODERN MATHEMATICS 135 binatorial analysis from the domain of mathematical recreations into that of severely practical mathematics; and only about fifty years elapsed between the creation of the mathematical theory of probability and the calculation of mortality tables by its use. In modern combinatorial analysis the calculus is indis- pensable in obtaining usable approximations to formulas beyond practicable exact computation. 1 The other great advance in the discrete division, Fermat’s creation of the modern higher arithmetic, was for long restricted to the study of relations between subclasses of the class of all rational integers. Since about 1850, numerous arithmeticians have extended the classical theory of Fermat and his successors to vastly wider classes of integers. The contribution to all mathematics of the higher arithmetic has been indirect, in the invention of new techniques, particularly in modern higher algebra and to a lesser extent in analysis, primarily for applica- tion to problems concerning the rational integers. Conversely, extensive tracts of the modern theory of numbers would not exist had not analysis made them possible. The careers of synthetic projective geometry and symbolic logic afford an interesting contrast in mathematical obsolescence and survival. Both will be noted in subsequent chapters; here we remark only the striking difference between their fate and the uniform prosperity of the other creations of the seventeenth century. After its invention by Dcsargucs and Pascal, synthetic projective geometry languished till the early nineteenth century, when it became extremely popular among geometers with a distaste for analysis. Leibniz’ dream of a mathematical science of deduction lay dormant till the mid-nineteenth century, and even then it appealed to but very few, atthough Leibniz had foreseen the importance of symbolic logic for all mathematics, and had himself made notable progress toward an algebra of classes. Only in the second decade of the twentieth century did mathematical logic become a major division of mathematics. Concurrently, synthetic projective geometry was receding defi- nitely into the past with the reluctant admission that an essen- tially Greek technique, even when revitalized, is hopelessly impotent in competition with the analytic methods of Descartes and his successors. From all this it is clear that after the period of Archimedes, Euclid, and Apollonius, that of Descartes, Fermat, Newton, and Leibniz is the second great age of mathematics. If the funda- 136 THE DEVELOPMENT OF MATHEMATICS mental distinction between the old and the new can be suggested in a word, it may be said that the spirit of the old was synthesis that of the new, analysis. * Anticipations’ Before proceeding to the individual advances, we must dis- pose of a purely historical matter which will not be further discussed. It concerns numerous aborted or sterile ideas which have not passed into living mathematics. Behind each of the major advances of the seventeenth cen- tury were many short steps in the general direction of each, and some of these partial advances all but reached their unperceived goals. At least that is what we might be tempted to imagine now. Looking back on these efforts, some may be inclined in the generosity of their hearts to believe that without these halted steps final success would have been long delayed or unattained. In the specific instances of analytic geometry and the calculus, an examination of the mathematics — not the sentiments — involved has convinced a majority of professionals that the alleged anticipations are illusory. Especially is this the opinion of men who themselves create mathematics and who know from dis- concerting experience that hindsight sees much to which fore- sight was blind. In retrospect we can trace the evolution of analytic geometry, for example, back to Hipparchus, or even to the ancient Egyp- tians. Like every astronomer who has recorded the positions of the planets, Hipparchus used coordinates, in particular latitude and longitude. But the use of coordinates entitles nobody to priority in the invention of analytic geometry; nor does even an extensive use of graphs. As any intelligent beginner who has understood the first three weeks of a course in analytic geometry knows, analytic geometry and the use of coordinates in the plot- ting of graphs are a universe apart. Only in the sense that they preceded analytic geometry are such comparatively childish activities anticipations of that geometry. This also is the judg- ment of a majority of professional geometers, who probably are as competent as anyone in this matter. Wc must refer the reader elsewhere 3 for a detailed evaluation and rejection of the romantic claims that several early mathe- maticians, and in particular Apollonius, Nicole Oresme (four- teenth century), and Kepler, ‘anticipated’ Descartes and Fermat in their independent invention of analytic geometry. To preserve THE BEGINNING OF MODERN MATHEMATICS 137 the balance, ana to exhibit 2 clean-cut instance of the absolute zero to which so many scholarly differences of opinion in the history of mathematics add up, we cite another evaluation of these ‘anticipations’ in which exactly the opposite conclusion is ably upheld.' 4 Dozens more on either side might easily be mentioned; the two selected will suffice to orient whoever may be interested, and to start him on his own critical evaluations. In each of the major advances of the seventeenth century some definite step led from confusion to 2 new method. Thus Newton himself states what gave him a hint for the differential calculus: “Fermat’s way of drawing tangents.” But there is one ‘anticipator’ of the calculus, B. Cavalieri (1598-1647, Italian), who merits more than a passing citation for the lasting mischief his ‘anticipation’ has done. Cavalieri's method of indivisibles has endured, to the distraction of hun- dreds of teachers of the elementary calculus who must extirpate heretical notions of infinitesimals from their students’ minds. In the United States, much of this elementary confusion can be traced to a generation of college teachers who were thoroughly indoctrinated in their school course in solid geometry with Cavalieri’s method of indivisibles. Their school geometries contained a seductive section on what some textbook writers called Cavalieri bodies: and these indivisibly-divisible non- entities were used, among other absurdities, to inculcate a disas- trously nonsensical account of mensuration in three dimensions. Cavalieri did not anticipate the calculus: he committed the unpardonable sin against it. But for his indivisibles and their absorption by scores of otherwise rational men who were to become college teachers, the common delusion that an infini- tesimal is a ‘little zero’ would have been extinct two generations ago. The historical appeal of Cavalieri’s indivisibles is undeniable, and that, perhaps, is why some historians palliate their flagrant offenses. They were inspired by the scholastic lucubrations of Thomas Bradwardine, Archbishop o? Canterbury (thirteenth century), and the submathcmatical analysis of Thomas Aquinas. As Cavalieri never defines his indivisibles explicitly, it is open to his apologists to read into them anything they know should be there but is not. But if his mystical exposition (1635) means anything at all. Cavalieri regarded a line as being composed of points like a string of countable but dimensionless beads, a surface as made un simiiarlv of lines without breadth, and a solid 138 THE DEVELOPMENT OF MATHEMATICS as a stack of surfaces without ultimate thickness. These are the very notions which a conscientious teacher will purge out of his students if it takes four years. A historical argument in favor of these indivisibles is that Leibniz was acquainted with them. But even this does not make Cavalieri an anticipator of the calculus. As will appear shortly, Newton clearly recognized the untenability of indivisibles; and although he did not fully succeed in clearing up his own diffi- culties, he did not mistake nonsense for sound reasoning. That is the fundamental distinction between one who imagined the calculus and one who did not. Contrary estimates of Cavalieri’s work are readily available. Descartes , Fermat , and analytic geometry Rene Descartes (1596-1650, French) is more widely known as a philosopher than as a mathematician, although his philosophy has been controverted while his mathematics has not. Descartes’ family was of the lesser French nobility. His mother died shortly after her son’s birth, but an unusually humane father and a capable nurse made up for this loss. After a broad education in the humanities at the Jesuit college of La Fleche, Descartes lived for two years in Paris, where he studied mathematics by himself, before joining Prince Maurice of Orange at Breda as a gentleman officer in 1617. In 1621, Descartes aban- doned his military career, partly because he had seen enough service both active and passive, partly because, as he declared, three dreams on the night of November 10, 1619, suggested the germs of his philosophy and analytic geometry. Much of the remainder of his life was spent in Holland, where he was safer from possible religious persecution than he would have been in France. These were his productive years; and in spite of his desire'tfor tranquillity, he could not conceal the greatness of his thowht. Rumors of what he was thinking were discussed wherever cshers with minds akin to his own dared to think. Largely througVf he efforts of F ather M. Mersenne (1588— 1648, French) of Pariycwho acted as intermediary between the French intellectuals ail«<the justly cautious Descartes, his fame spread over all Europei’t In 1637 Descartes published the work on which his greatness as a mathematician rests, the Discours de la methode four Men conduire sa raison et chercher la verite dans les sciences , the third THE BEGINNING OF MODERN MATHEMATICS 139 and last appendix of which, La gcomclrie , contains his subversive invention. The closing months of his life were spent as tutor to the young and headstrong Queen Christina of Sweden. The rigors of a Stockholm winter and the inconsiderate demands of his royal pupil caused his death. In accordance with the ideals of his age, when experimental science was first seriously challenging arro- gant speculation, Descartes set greater store by his philosophy than his mathematics. But he fully appreciated the power of his new method in geometry. In a letter of 1637 to Mersenne, after saying “I do not enjoy speaking in praise of myself,” Descartes continues: "... what I have given in the second book on the nature and properties of curved lines, and the method of examin- ing them, is, it seems to me, as far beyond the treatment in the ordinary geometry, as the rhetoric of Cicero is beyond the a, b, c of children.” 6 The famous appendix 6 on geometry consists of three books, of which the second is the most important. The third is devoted mostly to algebra. It will suffice to restate in current terminology the essential features of Descartes’ advance.® A plane curve is defined by some specific property which holds for each and every point on the curve. For example, a circle is the plane locus of a point whose distance from a fixed point is constant. Any point on a curve is uniquely determined by its coordinates x and y; and an equation /(.v, y) = 0 between the coordinates completely represents the curve when the specific geometric property defining the curve is translated into a rela- tion, denoted by the function /, between the coordinates x, y of the particular-general point on the curve. There is thus established a one-one correspondence between plane curves and equations in two variables x } y: for each curve there is a definite equation /(.v, y) = 0, and for each equation /(.V, y) = 0 there is a definite curve. Further, there is a similar correspondence between the alge- braic and analytic properties of the equation /(.v, y) = 0 and the geometric properties of the curve. Geometry is thus reduced to algebra and analysis. Conversely, analysis may be spoken in the language of geometry, and this has been a fecund source of progress in analysis and mathematical physics. The implications of Descartes’ analytic reformulation of geometry are obvious. Not only did the new method make possi- 140 THE DEVELOPMENT OF MATHEMATICS ble a systematic investigation of known curves, but, what is of infinitely deeper significance, it potentially created a whole universe of geometric forms beyond conception by the synthetic method. Descartes also saw that his method applies equally well to surfaces, the correspondence here being between surfaces defined geometrically and equations in three variables. But he did not develop this. With the extension to surfaces, there was no reason why geometry should stop with equations in three vari- ables; and the generalization to systems of equations in any finite number of variables was readily made in the nineteenth century. Finally, in the twentieth century, the farthest extension possible in this direction led to spaces of a non-denumerable infinity of dimensions. The last are not mere fantasies of the mathematical imagination; they are extremely useful frameworks for much of the intricate analysis of modern physics. The path from Descartes to the creators of higher space is straight and clear; the remarkable thing is that it was not traveled earlier than it was. Another direct road from Descartes to the present may be noted in passing. The formula (#1 — # 2 ) 2 + (yi — y 2 ) 2 for the square of the distance between any points (xi, yi), (x 2 , y 2 ) in a plane (surface of zero curvature) suggested the corresponding formulas in differential geometry for the square of the line element joining neighboring points in any space, flat or curved, of any number of dimensions, as quadratic differential forms. The germ of this long evolution was the Pythagorean theorem. In details, Descartes’ presentation differs from that now current. Thus, he used only an ar-axis and did not refer to a y-axis. For each value of X he computed the corresponding y from the equation, thus getting the coordinates x and y. The use of two axes obviously is not a necessity but a convenience. In our terminology, he used the equivalents of both rectangular and oblique axes. But in one important particular his procedure was needlessly restricted. He considered equations only in the first quadrant, as it was thence that he translated the geometry into algebra. This consistent but unnecessary limitation led to inex- plicable anomalies in the translation back from algebra to geometry. As analytic geometry evolved and negative numbers were fearlessly used, the restriction was removed. By 1748, when Euler codified and extended the work of his predecessors, both plane and solid analytic geometry were practically perfected, except for the introduction of homogeneous coordinates in 1827. THE BEGINNING OF MODERN MATHEMATICS 141 The new method was not fully appreciated by Descartes’ contemporaries, partly because he had deliberately adopted a rather crabbed style- When geometers did see what analytic geometry meant, it developed with great rapidity. But it was only with the invention of the calculus that analytic geometry came into its own. As early as 1704 Newton 7 was able to classify all plane cubic curves into seventy-eight species, of which he exhibited all but six. This comparatively early work in the geometry of higher plane curves is especially remarkable for its discussion of the nature of the curves at infinity, and for New- ton’s assertion, which he did not elucidate, that all species are obtainable as projections (‘shadows’) of the curves y- = ax z + bx- 4- cx 4- d. When we reflect that, only sixty-seven years before Newton published this work, geometers had been labori- ously anatomizing those other ‘shadows, 5 the conic sections, by the synthetic method of Apollonius and had not even imagined Newton’s cubics, we begin to appreciate the magnitude of the revolution Descartes precipitated in geometry. It is evident from Descartes’ explanation of his method that he had an intuitive grasp of the elusive concepts ‘variable’ and ‘function,’ both of which are basic in analysis. Moreover, he intuited continuous variation. Vieta before him had used letters to denote arbitrary constant numbers; Descartes knew that the letters in his equations represented variables, and he clearly recognized the distinction between variables and arbitrary constants, although he defined neither formally. The significance of this advance for the calculus that was to follow only sixteen years after his death is plain. Descartes’ progress in generality is illustrated by two of his minor but geometrically important observations. He classified algebraic curves according to their degrees, and recognized that the points of intersection of two curves are given by solving their equations simultaneously. The last implies what actually is a major advance over all who had previously used coordinates: Descartes saw that an infinity of distinct curves can be referred to one system of coordinates. In this particular he was far ahead of Fermat, who, apparently, overlooked this crucial fact. Fermat may have taken it for granted, but nothing in his work shows unequivocally that he did. Still seeking generality, Descartes separated all curves into two classes, the “geometrical” and the “mechanical.” This is curious rather than illuminating. He defined a curve to be 142 THE DEVELOPMENT OF MATHEMATICS geometrical or mechanical according as (in our terminology) dy/dx is an algebraic or a transcendental function. Although this classification was abandoned long ago, it affords an interesting sidelight on the quality of Descartes’ mind. The current defini- tion of a transcendental curve as one which intersects some straight line in an infinity of points was given by Newton in his work on cubics. Descartes’ method for finding tangents and normals need not be described, as it was not a happy inspiration. It was quickly superseded by that of Fermat as amplified by Newton. Fermat’s method amounts to obtaining a tangent as the limiting position of a secant, precisely as is done in the calculus today. The his- torical significance of this conception is evident when it is recalled that the tangent at ( x , y ) is drawn by a simple Euclidean con- struction once its slope dy/dx is known. Fermat’s method of tangents is the basis of the claim that he anticipated Newton in the invention of the differential calculus. It was also the occasion of a protracted controversy with Descartes. We pass on to Fermat and his part in the invention of ana- lytic geometry. There is now no doubt that he preceded Des- cartes. But as his work of about 1629 was not communicated to others until 1636, and was published posthumously only in 1679, it could not possibly have influenced Descartes in his own inven- tion, and Fermat never hinted that it had. Fermat was one of those comparatively rare geniuses of the first rank, like Newton and Gauss, who find all their reward in scientific work itself and none in publicity. Under modern economic conditions, it is inexpedient for a scientist to hide his light under a bushel unless he wishes to starve to death in the dark, and few do. Of course it is impossible to say what effect the prospect of jobless starvation would have had on the more aloof scientists of the past. Some of them lived in economic security independently of any scientific work they might or might not do. Today men make the only livings they have at science or mathe- matics, and it seems like a misapplication of a warped yardstick to measure their professional ethics by those of a hypothetical past that may never have existed. For it has yet to be proved that a full mind can outargue an empty stomach. In Fermat s case, either lifelong security or excessive modesty made publica- tion of very minor importance to him, and as a result his superb talents were all but buried in his own generation. Descartes, not Fermat, was the geometer whom others followed. THE BEGINNING OF MODERN MATHEMATICS 1 43 Pierre de Fermat (1601-1665, French, date of birth disputed) cultivated mathematics as a hobby. His profession, like Vieta’s, was the law. As a counselor of the local parliament at Toulouse, he lived a quiet, orderly life which left him ample leisure for his favorite study. An accomplished linguist and classicist as well as a first-rank mathematician, Fermat knew the master- pieces of Greek mathematics at first hand. Fermat did not discover his extraordinary powers in mathe- matics till he was about thirty, and even then he seems scarcely to have realized their magnitude. From his letters we get the impression that he regarded himself as a rather ingenious fellow, capable occasionally of doing a little better than Apollonius and Diophantus, but not very much after all in comparison with the ancient masters. Such sincere modesty would be engaging were it not exasperating: arithmeticians today would give a good deal for a glance at the methods Fermat must have devised but never published. In partial compensation for his indifference to publi- cation, Fermat was a voluminous correspondent. With the exception already noted concerning the use of one coordinate system for the representation of any number of curves, Fermat’s analytic geometry 8 appears to be as general as that of Descartes. It is also more complete and systematic. 9 By 1629, according to Fermat’s own dating, which there is no reason to question, he had found the general equation of a straight line, the equation of a circle with center at the origin, and equa- tions of an ellipse, a parabola, and a rectangular hyperbola, the last referred to the asymptotes as axes. The year (1638) after Descartes had published his geometry, Fermat communicated to him the accepted method of finding tangents. This originated in Fermat’s investigation of maxima and minima, which he approached in substantially the same way as is done today in the calculus. What he did amounts to equat- ing the derivative f'(x ) of f(x) to zero to find the values of x which maximize or minimize f(x). Geometrically, this is equiva- lent to finding the abscissas of the points on the curve y = /(*) at which the tangent is parallel to the .r-axis. He did not proceed to higher derivatives or their geometrical equivalent to determine whether f(x) = 0 actually gives maxima or minima, as is necessary in a complete discussion. Nor did he isolate the calcula- tion of the derivative from its implicit occurrence in problems of maxima and minima. Descartes either did not grasp the supe- riority of Fermat’s method or was too chagrined to admit it, and 144 THE DEVELOPMENT OF MATHEMATICS liis side of the controversy over tangents became somewhat acrimonious. One positive gain has survived from this work on maxima and minima, Fermat’s principle of least time 10 in optics. 11 This was the first (1657, 1661) of the great variational principles of the physical sciences. As we shall pass on presently to Newton and his calculus, we may consider briefly here what the account given above of Fermat’s tangents implies. If accepted at its full value, it makes Fermat an inventor of the differential calculus. The greatest mathematician of the eighteenth century, Lagrange, did so accept it. But the verdict is not unanimous. The difference of opinion seems to hinge on Fermat’s implicit conception of initially ‘neighboring’ but ultimately coincident points on a curve. To maximize or minimize/^) Fermat replaced x by x -f- E, where E differs but little from zero. He then equated f(x) to f(x + E ), simplified the algebra, divided by E, and finally set E equal to zero. 12 If this is legitimate differential calculus, then Fermat in- vented that calculus. If it is not, it seems no more illegitimate than its historical rivals. Thus Newton in his exposition of 1704, discussing the “fluxion” of x n , n an arbitrary rational number, used his binomial formula (1676) to expand ( x + o) n , and formed the difference (a: + o) n — x n . He then said, “Now let these augments [namely, ( x + o) n — x n and o] vanish, and their ulti- mate ratio will be 1 to nx n_1 .” This was his method of “prime and ultimate ratios.” In the Leibnizian notation used today, Newton thus finds dx n /dx = nx n ~ x . The conclusion seems to be that either nobody in the seven- teenth century invented the differential calculus, or Fermat was one of those who did. The matter is not settled by citing New- ton’s conception of a limit, because he did not develop a theory of limits in what he actually printed. But on this debatable differ- ence of opinion everyone must form his own opinion after understanding the evidence as he may. Before leaving the creators of analytic geometry, we may mention three further items from Descartes, although only the first is related to geometry. Descartes devised the notation x, xx, x z , x A , . . . for powers, and made the final break with the Greek tradition of admitting only first, second, and third powers (‘lengths,’ ‘areas,’ ‘volumes’) in geometry. After Descartes, geometers freely used powers higher than the third without a THE BEGIN KING OF MODERN MATHEMATICS 145 qualm, recognizing that representability as figures in Euclidean space for all of the terms in an equation is irrelevant to the geometrical interpretation of the analysis. 13 The principle of undetermined coefficients was also stated by Descartes. Anything approaching what would now be admitted as a proof was about two centuries beyond the mathematics of his time. A second outstanding addition to algebra was the famous rule of signs given in even' text on the theory of equa- tions. This was the first universally applicable criterion for the nature of the roots of an algebraic equation. Even if it does not always yield any useful information, it admirably illustrates Descartes’ flair for generality which made him the mathe- matician he was. Newton, Leibniz, and the calculus In the history of Newton’s calculus, the temptation to read ‘anticipation’ into the works of his contemporaries and immedi- ate predecessors is perhaps stronger than in that of any other major advance in mathematics. Knowing what wc now do of the calculus and its implications in geometry and elementary kinematics, wc can look back on many isolated discoveries in those domains and see in them what we now recognize as steps toward differentiation. But the discoverers, sometimes to our amazement, completely missed what wc now perceive so plainly. They failed in each instance to take the last gigantic stride that now seems to us but a short step; and to credit them with strides they might have taken but did not is sheer sentimental romanticism. As a relevant exercise in distinguishing between mathe- matical insight and facile prophecy after the fact, students of the calculus may wish to test their critical powers on the history' of the “differential triangle” of Isaac Barrow (1630-1677, English). This was somewhat in the manner of Fermat. Ignoring this, we shall adhere to the generally accepted tradition and proceed on the hypothesis that Newton in his calculus did some- thing new. Isaac Newton (1642-1727, English), the posthumous son of a yeoman farmer, was bom near Grantham, Lincolnshire, and passed his boyhood there. As a boy he was only passively inter- ested in his schooiwork until he suddenly woke up at the age of adolescence. Earlier, he had shown unmistakable promise of experimental genius in the mechanical toys he invented and 146 THE DEVELOPMENT OF MATHEMATICS made to amuse himself and his young friends. It is interesting that both Newton and Descartes were delicate in childhood, and therefore had time to think and develop their own personalities while rougher boys were reducing one another to a very common denominator. Both matured into sturdy men, Descartes through military training, Newton by the inherited toughness of his farmer forebears. After a desultory attempt to learn farming, Newton was sent to Trinity College, Cambridge, in 1661 (age nineteen). His undergraduate career, from all that is definitely known of it, was not particularly distinguished. Before going to Cambridge he had skimmed Euclid’s Elements , and is said to have dubbed it “a trivial book.” When, later, he understood Euclid’s purpose, he revised his hasty judgment. In his own work he refers to Euclid with evident respect. His baffling encounters with “very little quantities” made him appreciate at least the tenth book of the Elements. It should encourage intelligent beginners to know that Newton found analytic geometry difficult at a first reading. Perhaps fortunately for mathematics, Newton’s studies were interrupted in 1665-6 by the Great Plague, when the university closed. Newton returned home, but not to farm. Before he was twenty-four years of age he had imagined the fundamental ideas of his fluxions (calculus) and his law of universal gravitation. On returning in 1667 to Cambridge, Newton was elected a fellow of Trinity, and in 1669 succeeded Barrow, who resigned in his favor as Lucasian professor of mathematics. His first work to become known beyond the narrow circle of his intimate friends was in optics, beginning with his lectures of 1669. As we are interested here mainly in Newton’s calculus, we shall merely summarize the material circumstances of his career, full accounts of which are readily accessible. These also describe his epochal work in optics, which will not be discussed here as it belongs rather to physics than to mathematics. In 1672 (age thirty), Newton was elected to the Royal Society, and from 1703 till his death was its president. His Principia, universally estimated by competent judges to be the greatest contribution to science ever made by one man, was composed in 1684-6 at the instigation of the astronomer E. Hal- ley (1656-1742), at whose expense it was printed in 1687. In 1689, and again in HOT Newton was elected to represent Cambridge University in Parliament. He had no taste for de- THE BEGINNING OF MODERN MATHEMATICS 147 bating, but he took his duties seriously, and showed a fine cour- age in championing the University’s rights against the dictatorial meddling of King James the Second. At the age of fifty (1692) he suffered a severe illness and lost interest in scientific work, although he retained his unsurpassed intellectual powers to the end of his life. Partly by his own desire, partly at the insistence of friends who wished to sec him honored, Newton entered public life when he tired of science, and in 1696 was made warden of the mint. Having successfully directed the reform of the coinage, he was promoted to the mastership in 1699. In 1705 he was knighted by Queen Anne, and in 1727 he died. He is buried in Westminster Abbey. Newton’s excessive reluctance to publish his scientific work reflects certain aspects of his character. Although by no means a shy or timid man, Newton had a strong distaste for anything bordering on controversy. An unintelligent dispute over his work in optics at the beginning of his career taught him that scientific men arc not always so objective as they might be, even in science, and he retired within himself in astonished disgust. Nor was his notorious indifference to the survival of his scientific work affectation. But for the adroit coaxing and goad- ing of Halley, the Principia would probably never have been written. Newton himself esteemed the theological writings to which lie devoted the leisure of his later years far more highly than his science and mathematics. Again, in his work on light, Newton had proved himself one of the most acute experi- mentalists in the history of science; so it was but natural that he should spend much time and a considerable amount of money on what we should call alchemy, but what in his day was orthodox chemistry. It was the ironic misfortune of this hater of profitless dis- putes to be embroiled in the most disastrous mathematical con- troversy in history, when some of his busy friends inveigled him into insinuating that Leibniz had plagiarized his own form of the calculus. We shall not discuss this, but merely state that the almost universal opinion now is that Leibniz invented his calculus later than Newton and independently. It must be gratifying to Englishmen to recall that it was another English mathematician, that born nonconformist Augustus De Morgan (1S06-187 j), who first undertook a judicial examination of the dispute and obtained some measure of justice for Leibniz. THE BEGINNING OF MODERN MATHEMATICS 149 ability, and his lack of a feeling for tangible things occasionally betrayed him in science. Like Descartes, M who also went astray in science, Leibniz is probably most widely known today for his philosophy; but to a modern scientific mind his monads are as fantastically absurd 15 as Plato’s eternal ideas. He thought incessantly. His unresting curiosity was attracted by everything and distracted by nothing. Perhaps the world is fortunate that much of his intelligence was dissipated in one way or other in the pursuit of money and fugitive honors. As the reward for a revolutionary essay on the teaching of the law, Leibniz at the age of twenty-one was engaged by the Elector of Mainz as general agent and legal adviser. Most of his time thereafter was spent in travel on diplomatic missions for the Elector until the latter’s death in 1673. Leibniz then became librarian, historian, and political factotum for the Brunswick family at Hanover. During visits to France and England on political or diplo- matic missions, Leibniz met the leading French and English men of science, and in exchange for some of their ideas disclosed his own. One such trade was to prove profoundly significant in the development of the calculus. If we seek the origin of modern work in the foundations not only of analysis but of all mathe- matics, we need look no farther than the following incident. Until lie met the great Dutch physicist and mathematician Christian Huygens (1629-1695) in Paris in 1672, Leibniz had but little if any competence in what was then modern mathematics. Such firsthand mathematical knowledge as lie had was mostly Greek. Huygens enlightened him and undertook his mathe- matical education. Leibniz proved himself an exceedingly apt pupil. The two became good friends, corresponding till the death of Huygens in 1695. Leibniz begged Huygens for criticism of his projects and, naturally, got it. It is only a speculation, but from Leibniz’ ambitious char- acter and his philosophic propensity for solving the universe, it is conceivable that his daring project for a universal symbolic reasoning was fostered by a determination to beat Descartes at his own game. The philosophic Frenchman had reduced all geometry to a universal method; the more philosophic German would similarly reduce ali reasoning of whatever kind to a universal “characteristic” or, as would be said today, a symbolic mathematical science. Leibniz in 1679-80 confided his project to Huygens. sc The physicist was not impressed. 150 THE DEVELOPMENT OF MATHEMATICS By fatal mischance, Leibniz chose a trivial and singularly uninteresting geometrical problem to illustrate what he in- tended , 17 with the result that Huygens misunderstood the entire matter. He became somewhat polemical. This failure to see what Leibniz meant is the more remarkable as Huygens himself had a scientific vision which saw forests in spite of their innum- erable trees. Possibly he was antagonized by Leibniz’ boastful attitude. In his misunderstanding of what the ambitious phil- osopher-mathematician was trying to do, Huygens for once descended to the pedantries of captious criticism. At first glance it may seem that Leibniz’ attempts toward symbolic logic are irrelevant in the development of the calculus. Nothing could be farther from the fact. We shall see presently that Newton in his early encounters with continuity lost himself in the racecourse of Zeno, of whose paradoxes he perhaps had never heard. Subtly disguised but yet the same, these hoary difficulties have perplexed every mathematician, from Newton in the seventeenth century to Weierstrass in the nineteenth, who has sought not merely to obtain useful or interesting results by routine differentiations and integrations, but to understand the calculus itself. The calculus was difficult to Newton and Weierstrass; it is easy only to those who understand it too easily. The modern attack on the fundamental problems of con- tinuity has revealed the nature of the difficulties which baffled Newton, Leibniz, and the more thoughtful of their successors. It seems safe to say that without the mathematical logic which Leibniz advocated, and which he started to create, the critical work of the twentieth century on the foundations of analysis, and indeed of all mathematics, would have been humanly impossible. Leibniz imagined the project of a ‘calculus’ of deductive reasoning; and if his own steps toward it weje but short and hesitating, nevertheless it was his bold conception which encour- aged others to proceed. It seems rather late in* the day, therefore, to persist in seeing Leibniz the mathematician merely as a major satellite of Newton. / Historical tradition reiterates to weariless the undisputed fact that the ^-notation of Leibniz is vastly) superior to the dots of Newton. But if, with Gauss , 18 we believe {that in mathematics notions are more important than notations, ( we must place the emphasis elsewhere. The greatest work of Leibniz, from the standpoint of modern mathematics, is not hi's improvement of THE BEGINNING OF MODERN MA Til EM A TICS 151 the differential and integral calculus, great though that was, but his calculus of reasoning. He shines by his own light. Little need be said here of Leibniz’ career as a diplomat. To him is attributed that epitome of unstable equilibrium which later jugglers of destiny were to worship as the Balance of Power. In his diplomacy, Leibniz was neither more nor less unscrupulous than any of his famous successors in that dubious art. He was merely less incompetent than the majority. Unlike some of them, Leibniz did not succumb beneath a weight of honors heaped on him by grateful princes, but died neglected and forgotten by those whose petty fortunes he had made. When his employer departed from Hanover to become King George the First of England, Leibniz was discarded in the library to continue his history of the Brunswick family, surely a fitting occupation for one of the supreme intellects of all time. Only his secretary followed him to his grave. Such were the two mortals who finally created the calculus. Newton’s version of the calculus Newton’s first calculus, of 1665-6, seems to have been abstracted from intuitive ideas of motion. A curve was imagined as traced by the motion of a ‘flowing’ point. The ‘infinitely short’ path traced by the point in an ‘infinitely short’ time was called the "momentum” and this momentum divided by the infinitely short time was the "fluxion.” If the “flowing quan- tity” is .v, its fluxion is denoted by A. In our terminology, if .v is the function /(t) of the time t, x is dx/dt, the velocity at time !. Similarly, the fluxion of .v is x, our d-x/dt-] A is d 3 x/dl 3 , and so on. 15 Newton regarded our dxfdt as the actual ratio of two “in- finitely small quantities” in this first calculus. He had no approach to a limit that would be recognized today. The follow- ing extract from the Principia (1687) will indicate that Newton himself was dissatisfied with his own refinement of the method of fluxions. It is objected that thercis no ultimate ratioof evanescent quantities because the proportion (ratio] before the quantities have vanished is not ultimate; and, when they have vanished, is none. But by the same argument, it might as well be maintained, that there is no ultimate velocity of a body arriving at a certain place, when its motion is ended: because the velocity, before the body arrives at the place, is not its ultimate velocity; when it has arrived, is none. But the answer is easy . , . There is a limit, which the velocity at the end of the motion may attain, but cannot exceed. 152 TIIL DEFELOPMEKT OF MATHEMATICS This is what Zeno and the tortoise knew, and what neither of them succeeded in clarifying. It is no disparagement of Newton, to observe that the foregoing extract might have been written by Aristotle; indeed it bears a singular resemblance to Aristotle’s discussion” of the infinite, the continuous, motion, and Zeno’s paradoxes. A further observation of Newton’s recalls Eudoxus: “It may also be argued, that if the ultimate ratios of evanes- cent quantities are given, their ultimate magnitude will also be given; and so all quantities will consist of indivisibles, which is contrary to what Euclid has demonstrated concerning incom- mensurables, in the tenth book of his Elements .” Again compare with Aristotle. From the last it is clear that Newton’s understanding of Euclid was sharper than Cavalieri's. It also suggests that the difficulties of intelligent beginners with limits and continuity are not mere willful perversity. In his third attempt (1701), Newton returns to the attack on continuity, and transfers the central difficult)' to an unanalyzed “continued motion”: “I consider mathematical quantities in this place not as con- sisting of very small parts; but as described by a continued motion. Lines are described, and thereby generated not by the apposition of parts, but by the continued motion of points 33 * * • It was considerations of this kind, among others, that drove the analysts of the nineteenth century to desperation and im- pelled them to attempt a meaningful foundation for the calculus. In spite of his resolute abandonment of “very small parts,” Newton never quite circumvented the “very' small quantity’’ which persistently annoyed him. In the Principia (Bk. I, See. I, Lemma I), he started toward a theory of limits and continuity: “Quantities, and the ratios of quantities, which, in any finite time, tend continuously to equality; and before the end of that time, approach nearer to each other than by any given difference, become ultimately equal. r Leibniz’ version Leibniz for his part favored a species of differential, as that highly elusive concept is frequently misunderstood by practical engineers today. Thus, to find the differential o: xy, he sub- tracted xy from (x -j- dx') (y -f dy) and rejected dx dy because he considered it negligibly small in comparison with x dy ana TIIL BEGINNING OF MODERN MATHEMATICS 155 y dx — nil without sound justification. This gave him the correct result, d{xy) — x dy -f- y dx. On safer ground, lie introduced the current notation for derivatives and the integral sign, J, an elongated s from sutnma (sum). Both Leibniz and Newton were familiar with the funda- mental theorem of the calculus connecting integrals as sums with integrals as anti-derivatives. They also established the elementary formulas of the calculus. It is of interest that the correct result for the derivative of a product eluded Leibniz on his first attempt. Rigor; anticipations It is generally agreed that reasonably sound but not neces- sarily final ideas of limits, continuity, differentiation, and integration came only in the nineteenth and twentieth centuries, beginning with Cauchy in 1821 -3. This raises an extremely interesting question: how did the master analysts of the eight- eenth century — the Bernoullis, Euler, Lagrange, Laplace — contrive to get consistently right results in by far the greater part of their work in both pure and applied mathematics? What these great mathematicians mistook for valid reasoning at the very beginning of the calculus is now universally regarded as unsound. No short answer is possible; but history shows that frequently the essential, usable part of a mathematical doctrine is grasped intuitively long before any rational basis is provided for the doctrine itself. The creative mathematicians between Newton and Cauchy obtained mostly correct results — according to present standards — because, in spite of their ineffectual attempts to be logically rigorous, they had instinctively apprehended the self-consistent part of their mathematics. Just as no short answer can dispose of our predecessors’ good fortune, so none can dispose of ours. Like them, we consistently get meaningful results, although we realize that there is much obscurity in the foundations of our own analysis. It is now generally admitted that neither Cauchy nor his more rigorous successor Weicrstrass said the last word, and we may con- fidently expect that it will not be uttered in our generation. Whatever else may be said of Newton’s calculus, it is still true that he endowed mathematics and the exact sciences with their most effective method of exploration and discover}'. Linked to his own law of universal gravitation, the calculus in 154 THE DEVELOPMENT OF MATHEMATICS less than a century gave a more comprehensive understanding of the solar system than had accrued from thousands of years of prc-dynamical astronomy. And when differential equations, Newton’s method of inverse tangents, were applied to the physical sciences, a new and unsuspected universe was revealed. The experimental method of Galileo combined with the calculus of Newton and Leibniz generated modern physical science and its applications to technology. To conclude this account of the emergence of the calculus, wc shall compensate for the deliberate neglect of pseudo anticipa- tions by citing a real one in a subject which is basic for the modern attack on the foundations of analysis. Galileo observed as early as 1638 that there are precisely as many squares 1, 4, 9, 16, 25, ... as there are positive in- tegers altogether. This is evident from the sequences 1, 2, 3, 4, 5, 6,. . . , . . . I 2 , 2 2 , 3 2 , 4 2 , 5 2 , 6 2 , . . . , 71 2 , . . . He thus recognized the fundamental distinction between finite and infinite classes that became current in the late nineteenth century. An infinite class is one in which there is a one-one correspondence between the ■whole class and a subclass of the whole. Or, what is equivalent, there are as many things in some part of an infinite class as there are in the whole class. The like is not true of finite classes. A class whose elements can be put in one-one correspondence with the integers 1, 2, 3, . . . is said to be denumerable. All the points in any line segment, finite or infinite in length, form a non-denumerable set. A basic course in the calculus (usually the second) starts from the theory of point sets. The distinction between denumerable and non-denumerable classes was not stated by Galileo; it was observed about 1840 by Bolzano and in 1878 by Cantor. But Galileo’s recognition of the cardinal prop- erty of all infinite classes makes him one of the genuine anticipa- tors in the history of the calculus. The other was Archimedes. Emergence of the mathematical theory of probability Games of chance are probably as old as the human desire to get something for nothing; but their mathematical implica- tions rvere appreciated only after Fermat and Pascal in I6a4 reduced chance to law. Galileo had given a correct solution oi a gaming problem by laborious tabulation of possible cases, but THE BEGINNING OF MODERN MATHEMATICS 155 did not proceed to general principles. The “problem of points” which inspired the originators of the mathematical theory of probability was well known to Cardan who, among his other accomplishments, was a reckless gambler. He, however, did nothing of importance toward a science of chance; and it is customary without any quibbling to regard Pascal and Fermat as the founders of mathematical probability. In the epoch-making problem, the first of two players who scores n points wins. If the game is abandoned when one has made a points and the other b, in what ratio shall the stakes be divided between them? This reduces to calculating the prob- ability each has of winning when the game is stopped. It is assumed that the players have equal chances of making a point. The problem was proposed to Pascal by a highly intelligent gentleman addicted to gaming, Antoine Gombaud, Chevalier dc Mere, and Pascal communicated it to Fermat. Both solved it correctly, but by different reasoning. In some of his work, Pascal made a slip, which Fermat corrected. Thus originated the mathematics of chance which today is basic in all statistical analysis from stock-market trends and insurance to intelligence tests and biometrics. As modern physics has become more certainly uncertain, the mathematics of probabilities has steadily increased in scientific importance. Newtonian mechanics is applicable to a completely determinate science in which differential equations imply the future history of a mechanistically determined universe. For the scientific interpretation of laboratory experiments, particularly in atomic physics, the strictly mechanical method of Newton, Lagrange, Laplace, and their successors, originating in Galilean mechanics and dynamical astronomy, is no longer adequate and is being increasingly supplemented by the mathematics of statistics and probability. The necessary mathematics all developed from the fundamental principles of mathematical probability laid down by Fermat and Pascal in about three months by a painstaking application of uncommon sense. 21 Later applications of analysis to the theory of probability, chiefly to obtain usable approximations to the very large num- bers occurring in even simple combinatorial problems, have made the modern theory highly technical. But with the excep- tion of epistemological difficulties concerning the meaning of probability, the basic principles remain those of 1654 as stated in intermediate texts on algebra. In this connection it should be 156 THE DEVELOPMENT OF MATHEMATICS mentioned that the versatile Huygens got wind of what Pascal and Fermat were doing, and in 1654 published one of the earliest treatises on probability. 22 The concept of mathematical expecta- tion was his. The relative permanence of the mathematical foundations of probability as laid down in the seventeenth century is character- istic of the mathematics of the discrete, in which generally there has been less need for revision than in analysis. The origin of modern arithmetic We shall understand ‘arithmetic 5 in the sense of the Greek arithmetica. Equivalents are ‘the higher arithmetic 5 and ‘the theory of numbers, 5 also, unfortunately, the hybrid ‘number theory 5 with its Aryan adjective and adverb, ‘number theoretic 5 and ‘number theoretically. 5 Gauss, the foremost exponent of the classical theory after Fermat, preferred the simpler ‘arithmetic 5 or, at longest, ‘higher arithmetic. 5 Modern arithmetic began with Fermat, roughly in the period 1630-65. Significant as was Fermat’s work in other departments of mathematics, he is usually considered to have made his greatest and most personal contribution in arithmetic. This extensive division of mathematics differs from others in its lack of general methods. Even comprehensive theorems appear to be more difficult to devise than, say, in algebra or analysis. Thus, in algebra there is a complete theory of the solution of algebraic equations in one unknown; in fact there are two complete theories. In arithmetic the simplest corre- sponding problem is the solution.! in integers of equations in two unknowns with integer coefficients, and for this there is nothing approaching a complete theory,. Such progress as has been made since Fermat will be noted in £-1 later chapter. Many of Fermat’s discoveries were either recorded as marginal notes in his books 1 ! (the arithmetic in his copy of Bachet’s Diophantus) or were communicated, usually without proof, to correspondents. Somle of his theorems were proposed by him as challenge problems to the English mathematicians. For example, he demanded a proof that the only positive integer solution of x 1 + 2 = y 3 is x — 5, y = 3. It will suffice to state those two of Fermat’s discoveries which appear to have had the profoundest influence on arith- metic and algebra since his time, ‘'and the one general metho m arithmetic due to him. THE BEGINNING OF MODERN MATHEMATICS 157 Fermat stated that if ?! is a positive integer not divisible by the positive prime p, then n p_1 — 1 is divisible by p. The Chinese “seem to have known as early as 500 B.c.” s * the special case it *= 2. Any student of the theory of algebraic equations, or of modern algebra, or of arithmetic, will recall the frequent appear- ance of this fundamental theorem. The first published proof was Euler’s in 1738, discovered in 1732; Leibniz had obtained a proof before 1683 but did not publish it. The rule of priority in mathematics is first publication. The second famous assertion of Fermat, his celebrated ‘Last Theorem,* states that xF -f- y n — z n , xyz A- 0, n > 2, is im- possible in integers x, y, z, n. He claimed (1637) to have dis- covered a marvelous proof; and whether or not he had, no proof has yet (1945) been found. There seems to be but little point now in proving the theorem for special ?i’s, enough in that direc- tion being known to make it fairly plausible that the theorem is true. But, to take out insurance against a possible disproof tomorrow, it must be emphasized that arithmetic is the last place in mathematics where unsubstantiated guessing is either ethical or profitable. Numerical evidence counts for very little; 2 * the only luxury a reputable arithmetician allows himself is proof. It is generally agreed that the famous ‘Last Theorem,* true or false, is of but slight interest today. But its importance in the development of arithmetic and modern algebra ha3 been very great. This will be discussed in the proper place. Fermat’s general method, that of “infinite descent,” is pro- foundly ingenious, but has the disadvantage that it is often extremely difficult to apply. In the particular theorem for which Fermat invented the method, it is required to prove that every positive prime of the form 4« -f- 1 is a sum of two integer squares. From the assumption that the theorem is false for some such prime p, Fermat deduced that it is also false for a smaller prime of the same kind. Descending thus he proved, on the assumption of falsity, that 5 is not the sum of two squares. But 5 — 1 2 -f- 2 2 ; hence the theorem. The outstanding desideratum in arithmetic is the invention of general methods applicable to nontrivial types of problems. Further, “the arithmetical solution of a problem should consist in prescribing a finite number of purely arithmetical operations (exempt from all tentative processes), by which all the numbers satisfying the conditions of the problem, and those only arc 158 THE DEVELOPMENT OF MATHEMATICS obtained.” 1 * Nobody after Euclid and before Lagrange in the eighteenth century even distantly approached this ideal. Emergence of synthetic projective geometry The sudden rise of synthetic projective geometry in the seventeenth century appears now as a belated resurrection of the Greek spirit. As already noted, Pappus in the fourth century a.d. anticipated a cardinal property of cross ratios; and even earlier Menelaus (first century a.d.) may have proved a theorem which can now be interpreted similarly. But it was only with G. Desargues’ (1593-1662, French engineer and architect) eccentric Brouillon project (abbreviated title) of 1639 that synthetic projective geometry' was developed as a new and independent division of geometry. Doubtless the great advance in perspective drawing by the artists of the Renaissance made inevitable the emergence of a geometrical theory including perspective as a special case; and Desargues the architect was doubtless influenced by what in his day was surrealism. In any event, he composed more like an artist than a geometer, inventing the most outrageous technical jargon in mathematics for the enlightenment of himself and the mystification of his disciples. Fortunately, Dcsargucsian has long been a dead language. In current terminology, ‘projective’ means invariance under the group G of all general linear homo- geneous transformations in the space (of 1, 2, 3, . . . dimen- sions) concerned, but not under all the transformations of any group containing G as a subgroup. After his own fashion, Desargues discussed cross ratio; poles and polars; Kepler’s principle (1604) of continuity, in which a straight line is closed at infinity and parallels meet there; involutions; asymptotes as tangents at infinity; his famous theorem on triangles in perspective; and some of the projective properties of quadrilaterals inscribed in conics. Descartes greatly admired Desargues’ invention, but happily for the future of geometry did not hesitate on that account to advocate his own. Desargues’ most enthusiastic convert was the same Pascal who participated in the creation of the mathematical theory' of probability'. B. Pascal (1623-1662) was therefore a very con- siderable mathematician, even if, like Descartes and Leibniz, he is popularly remembered for other things. His magnitude as a religionist has overshadowed his accomplishments as a mathe- THE BEGINNING OF MODERN MATHEMATICS 159 matician and physicist, and for one who has ever heard of Pascal’s Essay pour Ies eoniques there must be a million who have read at least a page of his Pensees. If anything, Pascal was more genuinely precocious than Leibniz. As a boy he was no mere sponge absorbing the learning of others, but a creative mathe- matician. At twelve he rediscovered and proved for himself several of the simpler theorems of elementary geometry. Four years later he had composed the famous essay on conics, in which he developed the consequences of his hexagramma mysti- cum — pairs of opposite sides of a hexagon inscribed in a conic intersect in collinear points. Combining his mathematics and his talent for physics, Pascal at nineteen (16-12) invented an adding machine, the ancestor of all those in use today. This was greatly improved about thirty years later by Leibniz, whose machine did both addition and multiplication. Pascal made grateful acknowledgment to Desargues for his skill in projective geometry. Perhaps in all of his mathematics Pascal was the brilliant commentator rather than the bold originator. Organically and spiritually ill for most of his thirty- nine years, he was unable, apparently, to concentrate his powers on the creation of a comprehensive method in anything, and his brilliance was dispersed and dissipated in the piecemeal illumina- tion of other men’s ideas. Much of his mentality was absorbed in the religious controversies of his time and in hopeless attempts to reconcile his own internal conflicts. Beyond his ‘mystic hexagram’ and his share in probability, it cannot be said that Pascal’s contributions left more than a transient shadow on the surface of mathematics. The main stream flowed far deeper than he ever dreamed. Before quitting the field for about a century and a quarter, synthetic projective geometry' fought a terrific pitched battle to survive against its analytic antagonist, in the impressive Sec- liones co niece (1685) of P. de ia Hire (1610-1718, French). La Hire proved over three hundred projective theorems syn- thetically, and in an astounding appendix showed that all the theorems of Apollonius on conics arc obtainable by the method of projection. But even these spectacular gymnastics failed to convince geometers that the synthetic method is as supple as the analytic. The conics no doubt were idealized from the Archetypal Circle by Plato’s Geometer with the Eternal Idea of projective geometry’ at the back of his mind; but then, not all plane curves arc conics. Nor were they even when La Hire fought his desperate rearuunrd action npnin't. De* carte*’ analytic*. Syrtlwrir - r -„ jective geometry iap £ cd into ttmr^rr.rv cblivix;. ;• r. d - treat he* of Dc^nrpuc* and La Hire became emir :tor j ’ ra-hh-. The other incidental advance of the seventeenth t+r.uitv that likcwbe war forgotten for a re a* ‘in. the uni vernal char; Ltic of Leibniz, wiil be noted in a later chapter. We pr.", , t ) the oritrin of the applied mathematics which wa*- to dominate t; - work of Newton's most prominent successors for a cctuurv r:V* his death. Or:g:r. of rr.odrn s* * +• J-V **/ a The indebtedness of science and technology to pare rr.athr- m a tics was noted in the Prospectus. We shall now impret the other ride of the ledger at somewhat greater length than rruy seem necessary to those already conversant with the fact*. We do this because hypersensitive mathematician'* arc nuwttr.t: inclined to exalt unduly the freedom and purely unarT.rVAe charactcr of their creations, and to dwell exclusively on v r admitted indebtedness of science to mathematic:. Tim hb- lorical balance sheet indicates, as will be seen in greater sb-tril later, that science and modem mathematics are so clwriy nfhliatcd that neither owes the other anything, each borrow::,,- freely from the other and repaying its debt 1 - a hundredfold. Intermediate between pure and applied mathematic** Im- provements in numerical calculation arc of more imp .rtar.cr 5 * the applications than for mathematics itself. \s varkhnw, Lr example, accelerated the practical development* t.f ro’.ror. my but were not a necessity even to that most efficient servant >•: civilization. The pertinacity of a Kepler cannot be thwart' ; :y any amount of manual computation; and to claim that lv> rithms made modern astronomy or any other p c:<-nce p.‘ w c is to forget that human zeal — or ob'tinacy — in purs';:: o'. a fixed idea can withstand any finite punishment. But a' rithms undoubtedly hastened the sciences of the cmht nineteenth centuries on their way to whatever is :■ ultimate contribution to civilization, or to its dftruct:' must be included in any account of the origin of mathematics. The seventeenth-century invcntvn o* ;* v may therefore be properly assigned to applied mati.crr.a Modern applied mathematic* oririnated in Newt: ‘ of universal gravitation developed in his P'inr.r :e. before Newton wa* purclv descriptive. Tr.c tncv.cn-: t. THE BEGINNING OF MODERN MATHEMATICS 161 planets were described with increasing accuracy, and from the Babylonians to Ptolemy were fitted into geometrical frameworks of ever greater complexity. Copernicus simplified the geometry. But there was no physical hypothesis abstracted and con- solidated in postulates from which the geometry could be deduced. Before such postulates could be stated profitably, accurate observations to determine the facts were necessary. These were provided in abundance by Tycho Brahe (15*16-1601, Danish), whose industrious assistant for a short time, Johann Kepler, (1571-1630, German), subsumed the observations under the three laws of motion known by his name. The first two were published in 1609, the third in 1619: the orbit of a planet is an ellipse with the sun at one focus; the areas swept out in equal times by the line joining the sun to a planet are equal; the squares of the periodic times of the planets are proportional to the cubes of their mean distances from the sun. Kepler’s laws were the climax of thousands of years of an empirical geometry of the heavens. They were discovered as the result of about twenty-two years of incessant calculation, with- out logarithms, one promising guess after another being ruth- lessly discarded as it failed to meet the exacting demands of observational accuracy. Only Kepler’s Pythagorean faith in a discoverable mathematical harmony in nature sustained him. The story of his persistence in spite of persecution and domestic tragedies that would have broken an ordinary man is one of the most heroic in science. The contemporaneous invention of logarithms was to reduce all such inhuman labor as Kepler’s to more manageable proportions. The history of logarithms is another epic of per- severance second only to Kepler’s. Baron Napier of Merchistoun (1550-1617, Scotch), in the leisure remaining from his duties as a landlord and his unavailing labors to prove that the reigning pope was Antichrist, invented logarithms. When it is remembered that Napier died before Descartes introduced the notation nil, n, s ... for powers, we cease to wonder that it took him all of twenty years to reason out the existence and properties of logarithms. The fundamental idea of the correspondence bctv.-ccn two series of numbers, one in arithmetic, the other in geometric progression, . . . was explained by Napier through the conception of two points marine on separate straight lines, the one with uniform, the other with accelerated velocity. If the reader, with all his acquired modern knowledge, will attempt to obtain for hinswlf in this 162 THE DEVELOPMENT OF MATHEMATICS way a demonstration of the fundamental rules of logarithmic calculation he will rise from the exercise with an adequate conception of the penetrating genius of the inventor of logarithms. (G. Chrystal.) Add to this that Napier’s logarithm of n would be our 10 7 loge (10 7 ft" 1 ), where e is the base of the natural system. After the invention of the calculus, investigation of the logarithmic function, of greater significance in mathematics than the logarithms computed by its use, followed as a matter of course from the simple differential equation dy = y dx. Napier gave Tycho a forecast of his invention in 1594, and in 1614 published his Descriptio. In 1624 a usable table by H. Briggs (1561-1631, English) was published, as also was one by Kepler. Other tables quickly appeared, and by 1630 loga- rithms were in the equipment of every computing astronomer. For those interested in squabbles over priority, it may be recalled that logarithms are one of the most disorderly battle- grounds in mathematical history. It will suffice here to state the outcome of the fray as adjudicated in 1914. Napier’s priority in publication is undisputed; J. Biirgi (1552-1632, Prague) inde- pendently invented logarithms and constructed a table between 1603 and 1611, while “Napier worked on logarithms probably as early as 1594 . . . ; therefore, Napier began working on logarithms probably much earlier than Biirgi.” 26 The only facts concerning logarithms of any importance for the development of mathematics are those stated in the concluding sentence of the preceding paragraph. Disputes like this and the other over the calculus have made more than one man of science envy his successors of ten thousand years hence, to whom Newton and Leibniz, Napier and Biirgi, and scores of lesser contestants for individual fame will be semi- mythical figures as indistinct as Pythagoras. The harmonious geometry of Kepler’s laws challenged mathematical ingenuity to devise a hypothesis from which they could be deduced. Among others, Newton’s self-constituted rival and gadfly, the brilliantly original R. Hooke (1635-1703, English), had guessed and perhaps proved that Kepler’s laws implied an inverse-square law of attraction, but could not deter- mine the form of the orbit from this law. Newton, on being consulted in 1684, restored a proof, which he had discovered but mislaid, that the required orbit is an ellipse. This incident appears to have been the origin of the Principia. From his hypothesis of universal gravitation that any two particles o THE BEGINNING OF MODERN MATHEMATICS 163 matter in the universe, of masses mi, at a distance d apart, attract one another with a force proportional to m^m^/d- (m 1 , 7zt», d, and the force being measured in the appropriate units), Newton deduced Kepler’s laws. The deduction would have been impossible without a rational dynamics. This had been provided by Galileo 57 and by Newton himself. Just as the Pythagoreans had reduced the intuitive perception of form to geometry, and the great geo- metrical astronomers from Eudoxus and Hipparchus to Coper- nicus and Kepler had reduced the motions of the planets to geometry', so Galileo undertook the reduction of all motion to mathematics. He advanced beyond his predecessors chiefly because lie aided reason by experiment, determining the facts in connection with falling bodies by accurate, controlled observa- tion before venturing to mathcmaticize. To some it seems incredible that any human being could ever have believed it possible to reason out the behavior of falling bodies without appeal to experiment. But one of the greatest intellects in history, Aristotle, had sufficient confidence in his logic to legislate for a universe which has but little respect for the unaided intellect. Others 1 * see nothing questionable in attempts such as Aristotle’s, substituting for the classical and medieval belief in Aristotelian logic an eager faith in the creative power of the intricate tautologies of mathematics. It may yet be too early to judge which side, if either, is right; but it is a fact that Galilean science, not Aristotelian logic and metaphysics, made our material civilization what it is. Whether or not the legendary' experiment in which Galileo confounded the Aristotelian scholastics by dropping shot from the Leaning Tower of Pisa ever took place, 55 Galileo knew by 1591 that a one-pound shot and a ten-pound shot dropped simultaneously from the same height strike the ground simul- taneously. Experiments on motion down inclined planes gave him further data to be fitted into the mathematical theory' of motion he sought to construct. As tentative hypotheses were subjected to experimental verification, the cardinal definitions and postulates of dynamics began to emerge. In particular, Galileo mathcmaticizcd distance, time, veloc- ity, and acceleration into the scientific (experimentally measur- able) things they still are in classical dynamics. He sought to frame definitions that would respond to repeatable observations. He also understood an equivalent of Newton’s first postulate 164 THE DEVELOPMENT OF MATHEMATICS of motion — inertia: every body will continue in its state of rest or of uniform motion in a straight line, except in so far as it is compelled to change that state by impressed force. This postu- late contradicted the naive intuitions of Galileo’s predecessors and controverted the common sense of ages. Galileo also understood at least special cases of Newton’s second postulate: rate of change of momentum is proportional to the impressed force, and takes place in the direction in which the force acts. The mathematically important concept here is that of a rate; for rates are derivatives, and hence velocity, accelera- tion, and force are brought within range of the calculus. We have seen that Newton probably had velocity in mind when thinking about fluxions. “In the two plague years of 1665 and 1666,” as Newton states, he deduced from Kepler’s third law that “the forces which keep the Planets in their Orbs must be reciprocally as the squares of their distances from the centers about which they revolve: and therefore [I] compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the Earth, and found them answer pretty nearly.” 30 Further progress in the mathematical theory of gravitation was temporarily halted by Newton’s lack at the time of a theorem in the integral calculus: the gravitational attraction between two homogeneous spheres can be calculated as if the masses of the spheres were concentrated in their centers. Once this theorem is proved, the Newtonian law of universal gravita- tion is applicable. If there is one master key to dynamical astronomy, this is it. With this key Newton proceeded in 1685 to unlock the heavens. He also, for the first time, gave a rational theory of the tides. The Newtonian celestial mechanics was the first of the great syntheses of natural phenomena. From its very nature, celestial mechanics without the dynamics of Galileo and Newton, or without the calculus of Newton and Leibniz, is unthinkable. The Galilean method in science was to provide the model for even more recondite mathematical syntheses, as in the theories of heat, light, sound, and electricity. But the modern scientific method, invented by Galileo and Newton, of welding experi- ment and mathematics into a single implement of discovery and exploration was to remain fundamentally the same as it was in Galileo’s Discorsi and Newton’s Principia. TIIE BEGINNING OF MODERN MA THEM A TICS 165 In a day when science is being discredited by messianic ignoramuses with enormous followings, it is well occasionally to recall the cliche, trite though it may be, that without this union of experiment and mathematics our civilization would not exist. Less trite is the more recent observation that because of this very union our civilization may cease to exist. And, while we arc facing facts, we note the opinion of many observers that ever since the days of Aquinas science has been feared or secretly hated by nine human beings out of every ten who have sufficient animation to hate or fear anything. Science has been grudgingly tolerated since the days of Galileo and Newton only because it has increased material wealth. If science dies, mathematics dies with it. To give an indication of how significantly dynamics and the Newtonian theory have influenced analysis, we may cite a few specific instances, some of which will be considered more fully in later chapters. Since the earth is not a sphere but a spheroid, its attraction on an exterior mass-particle cannot be calculated with the same precision as if its mass were concentrated at the center. When astronomy became more exact after Newton, the slight departure from perfect sphericity had to be included in the calculations, and this necessitated the invention of new func- tions, such as Legendre’s in potential theory. So rudimentary a dynamical problem as Galileo’s of the time of vibration of a simple pendulum of constant length leads at once in the general ease to an elliptic integral. Such integrals, by inversion, gen- erated the vast theory of doubly periodic functions. These in their turn were recognized in the late nineteenth century as but special eases of automorphic functions, whose theory still is far from complete. All of the earlier functions together suggested to Lagrange, Cauchy, and others in the late eighteenth and early nineteenth centuries general theories of functions, culminating in the theory of functions of a complex variable. Fourier’s analytical theory of heat (final form 1822), devised in the Galileo-Newton tradi- tion of controlled observation plus mathematics, is the ultimate source of much modern work in the theory of functions of a real variable and in the critical examination of the foundations of mathematics. Finally, the gravitational interactions of a system of mass-particles, in particular of three, generated the theory of perturbations and all its intricate analysis: and the problem of 166 THE DEVELOPMENT OF MATHEMATICS three bodies, partly topologized in the late nineteenth century, is the source of the modern theory of periodic orbits from which a qualitative, topologized dynamics is rapidly developing. Geometry also has enriched itself by successive alliances with mechanics. In the seventeenth century the astronomical need for the accurate measurement of time inspired Huygens to construct the first pendulum clock (1656). Incidentally, he was compelled to investigate the (small) oscillations of a compound pendulum, the first dynamical problem beyond the dynamics of particles to be discussed mathematically. From practical clock making Huygens tvas led to his great work 31 in horology (1673), in which he defined and investigated evolutes and involutes. The cycloid, sometimes called the Helen of geometry, partly on account of its graceful form and beautiful properties, figures prominently in this science. Huygens proved the remarkable theorem that the cycloid is the tautochrone. In more recent times, the four-dimensional geometry of Pliicker (nineteenth century) in which straight lines instead of points are taken as the irreducible elements of space, found a ready interpretation in the dynamics of rigid bodies. Conversely, this dynamics sug- gested much to be done in line geometry. But what is perhaps the greatest service a physical science has ever rendered geom- etry was the sudden acceleration imparted to differential geometry by Einstein’s general relativity and his relativistic theory of gravitation in the second decade of the twentieth century. In the curiously sanguine months after the buzzards of Versailles had completed their labors, it was frequently said that Einstein’s work would outlast the memory of the world war, as the science and mathematics of Archimedes has outlasted the Punic wars in the consciousness of all but professional his- torians. Twenty years later the human race had made a complete recovery from its attack of optimism. Here we leave the mathematics of the seventeenth century and commit ourselves to ■'che turbulent stream that gushed from that inexhaustible sou reek CHAPTER S Extensions of Number In following the development of mathematics since the death of Newton (1727), we might start from any one of arithmetic, algebra, geometry, or applied mathematics. As arithmetic preceded the others in the historical order from Babylon to Gottingen, we shall discuss it first. Those who arc more inter- ested in one of the other topics may pass at once to it. The detailed growth of the number concept to be described in this and the following five chapters being quite intricate, we shall indicate first the principal features to be observed. Four critical periods About four centuries of generalisation, confused and hesi- tating at first, produced the number systems of analysis, algebra, mathematical physics, and the higher arithmetic of the twen- tieth century. The final gain left mathematics with three major acquisitions: the ordinary complex numbers of algebra and analysis, and their subclasses of algebraic integers; the hyper- complex number systems of algebra, geometry, and physics; the continuum of real numbers as it appears in the modern theories of functions of real and complete variables. The five periods of most radical change were the decade bisected by the year 1SD0, the late lS30*s and early lS-10's, the lS70*s, and the twenty years bisected by 1900. With the first period is associated one beginning of modern abstract arithmetic and algebra, in the use by Gauss (1 SOI ) of a particular equivalence relation, which he called congruence, to map an infinite class of integers on a finite subclass. The general method of mapping (homomorphism) implicit in this early K~ 168 THE DEVELOPMENT OF MATHEMATICS work was not clearly formulated and isolated for independent study until the twentieth century, when it became basic in abstract algebra, topology, and elsewhere. In the 1830’s, the British algebraists clearly recognized the purely abstract and formal character of elementary algebra. This was followed, in the 1840’s, by Hamilton’s quaternions and the vastly more general algebras of Grassmann, from which the vector algebras of mathematical physics evolved. From the standpoint of pure mathematics, the lasting residue of this period was a widely generalized conception of number. The 1870’s saw the inception of the modern attack on the real number system in the work of Cantor, Dedekind, Meray, and Weierstrass. The outcome, in the late nineteenth century, was the arithmetization of analysis and the beginning of the modern critical movement. What now appears as the most enduring residue of this stormy period is the enormous expansion of mathematical logic during the first four decades of the twen- tieth century. The third period passed into the fourth about 1897 with the first appearance of the modern paradoxes of the infinite. The latter were largely responsible for the sudden growth of mathe- matical logic, which has reacted strongly on all mathematics and in particular on the number concept. We shall have occasion to refer frequently in anticipation to problems as yet unsettled concerning the nature of number and the continuum of real numbers. Because a problem defies solu- tion is no ground for believing it to be unsolvable. The outstand- ing obstacles that have hitherto blocked clear perception of the nature of number may be removed tomorrow. In any event, none of the unsolved problems of number has stopped progress in both pure and applied mathematics. On the contrary, in pure mathematics these unresolved difficulties have inspired much valuable work; and for applied mathematics, even the most serious doubts have as yet proved wholly irrelevant in obtaining scientific conclusions which can be checked against experience in the laboratory. Having followed the development of number in its purely mathematical aspects, we shall return in a later chapter to the impact of science on mathematics. It will be seen that applied mathematicians have been justified in their bold use of an analysis which may not yet meet all the demands of logica rigor. EXTENSIONS OF NUMBER 169 From time to time we shall cal! attention to points of special significance to be observed while following details. A general observation may be emphasized here. As mathematics passed the year 1800 and entered the recent period, there was a steady trend toward increasing abstractness and generality. By the middle of the nineteenth century, the spirit of mathematics had changed so profoundly that even the leading mathematicians of the eight- eenth century, could they have witnessed the outcome of half a century’s progress, would scarcely have recognized it as mathe- matics. The older point of view of course persisted, but it was no longer that of the men who were creating new mathematics. Another quarter of a century, and it had become almost a dis- grace for a first-rank mathematician to attack a special problem of the kind that would have engaged Euler in much of his work. Abstractness and generality, directed to the creation of universal methods and inclusive theories, became the order of the day. There had been one precedent in the eighteenth century' for such work, the dynamics of Lagrange. There is a second clue through the intricate development. This leads back to Pythagoras, and is so suggestive that we shall describe it next by itself in relation to the preceding sketch. The Pythagorean adventure Perhaps the feature of greatest general interest in the entire development is the wide departure from the Pythagorean pro- gram of basing all mathematics on the ‘natural* numbers 1, 2, 3, . . . in the periods of intenscst creativity, and the final return to Pythagoras for a brief interval after the natural num- bers had been extended to meet the demands of analysis, geom- etry, physics, algebra, and the higher arithmetic. What was probably the golden age of the Pythagorean program lasted through the second half of the nineteenth century. Thereafter, the modern critical movement sought to base the natural num- bers, and hence all their acquired extensions, on mathematical logic. This later program was already strongly hinted in that of the nineteenth century', which had attempted to derive num- ber from the theory' of infinite classes. To be appreciated as the great adventure in thought that it is, this circling movement away from Py'thagoras and back to him, with the subsequent firing off at a tangent he was incapable of imagining, must be inspected in some detail. This we shall do in the following five chapters. We shall see modern mathematics 170 THE DEVELOPMENT OF MATHEMATICS as a whole becoming increasingly self-conscious and critical of its own naive behavior in the eighteenth century, and passing safely through adolescence in the decade 1820-30. Mathematics then became less interested in the uncritical analysis that produced surprisingly accurate results in the heavens with Laplace’s celestial mechanics (1799, 1802, 1805), and on earth in the intuitive calculations of Fourier’s analytic theory of heat conduction (1822). Most of the great mathematicians of the eighteenth and early nineteenth centuries were more like engi- neers than modern mathematicians in their thinking; a formula revealed in a flash of intuition, or hastily inferred from loose reasoning, was as good as any other provided it worked. Their formulas worked admirably. Gauss (1777-1855) was the first great mathematician to rebel successfully against intuition in analysis. Lagrange (1736-1813) had tried and failed. The focus of the last serious trouble was found most unex- pectedly in the speciously innocuous natural numbers 1, 2, 3, . . . that, since the days of Pythagoras, had been eagerly ac- cepted by mathematics as manna from heaven. Indeed L. Kronecker (1823-1891, German), himself a confirmed Pytha- gorean and one of the leading algebraists and arithmeticians of the nineteenth century, confidently asserted that “God made the integers; all the rest is the work of man.” By 1910, some of the more wary mathematicians were inclined to regard the natural numbers as the most effective net ever invented by the devil to snare unsuspecting men. Others, of a yet more mystical sect, maintained that the natural numbers have nothing super- natural of either kind about them, asserting that the ‘unending sequence’ 1, 2, 3, . . . is the one trustworthy ‘intuition’ vouch- safed to Rousseau’s natural man. The tribes of the Amazon Basin were not consulted. Clashes between these and other opposing factions of mathe- matical orthodoxy temporarily relegated the Pythagorean program to limbo shortly after 1900. All parties to the many- sided dispute united in torturing logic into new and fantastic shapes to make it reveal at last what meaning, if any, there may be in the natural numbers and in the dream of Pythagoras con- cerning them. Do these numbers tell the truth about mathe- matics and nature, or do they not, ‘truth’ being merely a self-consistent description ? If they do not, is it necessary for human needs that the mathematics constructed on the natural numbers be ‘true’ in this sense? Whatever may be the answer EXTENSIONS OF NUMBER 171 to the first question, that to the second seems to be an emphatic No. A vast amount of mathematical reasoning now known to be unsound led in the past to extremely useful consequences. However, our concern here is not with such profound questions, all of which may be meaningless, but with the technical mathe- matics which has bred these inquiries and many more like them. But we may note in passing that mathematics is not the static and lumpish graven image of changeless perfection that some adoring worshippers have proclaimed it to be. The extensions of the number system since the sixteenth century are one of the outstanding accessions of all mathematics. In the opinion of those competent to estimate the technical evidence, these extensions arc likely to be of value for many years to come. The ‘crisis’ of the early twentieth century in all mathematics, induced by an uncritical acceptance of the formal- ism that had generated the successive extensions of the number system, was precipitated by too bold a use of infinite classes in an attempt to be logically rigorous. Infinite classes penetrated the domain of number, as will appear later, from two diametrically opposite points, namely, the finite cardinal numbers of common arithmetic and the continuum of analysis. In the higher arith- metic also, Dcdckind’s generalization (about 1870) of the rational integers and their unique decomposition into primes to the algebraic integers with a corresponding unique decomposi- tion into prime ideals introduced dcnumerably infinite classes of algebraic integers. The concurrent entry of non-denumerably infinite classes of rational numbers came with the theories of Cantor and Dedckind, devised to provide the continuum of real numbers in analysis with a self-consistent foundation. Thus the central obstacle, the mathematics of the infinite, that had stopped Pythagoras halted his successors over two thousand years after he had become a legend. Eudoxus seemed to have discovered a way round the obstacle, or perhaps through the thickest part of it; and the builders of the modern continuum, following essentially the same path, sought only to clear it of obstructions and give it a firmer foundation. What at first looked like security appeared on closer inspection to be an illusion. The road had yet to be built. To anticipate the report in a later chapter and bring this forecast to a close, we recall the singular conversion of H. Poincare (1S54-19I2, I” rench) in 190S. This, in a way, sums up the progress of 2,300 years. At the close of the nineteenth cen- 172 THE DEVELOPMENT OF MATHEMATICS tury, Poincare was a major prophet of a self-confident mathe- matics. In 1900 he declared that all obscurity had at last been dispelled from the continuum of analysis by the nineteenth- century philosophies of number based on the theory of infinite classes {Mengenlehre). All mathematics, he declared, had finally been referred to the natural numbers and the syllogisms of tradi- tional logic; the Pythagorean dream had been realized. Hence- forth, reassured by Poincare, timid mathematicians might proceed boldly, confident that the foundation under their feet was absolutely sound. Eight stormy and eventful years changed the prophet’s vision: “Later generations will regard the Mengenlehre as a disease from which one has recovered.” Thirty years after Poincare’s somewhat caustic prognosis was pronounced, the theory from which mathematics was to have recovered was still flourishing. This of course disproves nothing; Euclid’s geometry lasted unmodified in the minds of generation after generation of mathematicians who for two thousand years believed it to be flawless. Both the prognosis and its possibly retarded realization are recalled here merely to exhibit in a just perspective the great acquisitions of number since the sixteenth century. If in its continual development mathematics seldom if ever attains a finality, the constant growth does mature some residue that persists. But it is idle to pretend that what was good enough for our fathers in mathematics is good enough for us, or to insist that what satisfies our generation must satisfy the next. Extension by inversion and formalism The earliest extensions of the system of natural numbers were the Babylonian and Egyptian fractions. These illustrate one prolific method of generating new numbers from those already accepted as understood, namely, inversion. To solve the problem ‘by what must 6 be multiplied to produce 2?’, a new kind of ‘number,’ the fraction must be invented. Here the direct operation is multiplication, and the inverse, division. The other pairs of elementary inverses are addition and subtraction; raising to powers and extracting roots. All of these elementary operations were known to the an- cients. The inverses, division and subtraction, of the rationa operations, multiplication and addition, necessitated the mven tion of common fractions and negative numbers; the operation inverse to powering was in part responsible for the invention o EXTENSIONS OT NUMBER 173 irrationals, including the pure imaginaries and the ordinary complex numbers. The solution of an algebraic equation, or of a system of such equations in several unknowns, can be restated as a problem in inversion with respect to iterations of addition and multiplication. Up to about 1840, algebraic equations were probably the most prolific source of extensions of the natural numbers. Considering the extensions up to and including the acquisi- tion of ordinary complex numbers, we shall take a point of view which may be indefensible historically but which can be justified mathematically: mere accidental encounters with, say, negative numbers do not constitute mathematical discover}'. Nor docs a rejection of imaginary roots of equations entitle anyone to priority in the invention of complex numbers. Until a conscious attempt was made to understand negative and complex numbers, and to state rules, however crude, for their use wherever they might occur, neither had any more right to be considered a mathematical entity than has an unconceivcd child to be con- sidered a human being. Mathematically, these numbers did not exist until the conditions indicated were met. Professional historians are in substantial agreement on the following details in the development of negatives. Diophantus, in the third century A.D., encountering —4 as the formal solution of a linear equation, rejected it as absurd. In the first third of the seventh century, Brahmagupta is said to have stated the rules of signs in multiplication; he discarded a negative root of a quadratic. The rules of signs became common in India after their restatement by Mahavira in the ninth century. AI- Khowarizmi, of about the same time, made no advance except that he appears to have exhibited a positive and a negative root for a quadratic without explicitly rejecting the negative. Of the Europeans, Fibonacci in the early thirteenth century rejected negative roots, but took a step forward when he inter- preted a negative number in a problem concerning money as a loss instead of a gain. It has been claimed that the Indians did likewise. L. Pacioli (1445 r-l 514, Tuscan) in the second half of the fifteenth century is credited w\tV a knowledge of the rule of signs on such evidence as (7 — 4) (4 — 2) = 3 X 2 = 6. M. Stifcl 1 (14S7I-1567, German) a fine algebraist for his time, called negative numbers absurd in the middle of the sixteenth century. Cardan, in his Ats map: a (1545), stated the rule ‘minus times minus gives plus’ as an independent proposition; he also 174 THE DEVELOPMENT OF MATHEMATICS is said to have recognized negative numbers as ‘existent,’ but on evidence which seems doubtful. In fact, he called negatives ‘fictitious.’ Bombelli in 1572 showed that he understood the rules of addition in such instances as m — n , where m, n are positive integers. Vieta, about the same time, rejected negative roots. Finally J. Hudde (1628-1704, Dutch) in 1659 used a letter to denote a positive or a negative number indifferently. As an historical curiosity, it may be mentioned that T. Harriot (1560- 1621, English) w'as one of the first Europeans to duplicate the feat of the ancient Babylonians in permitting a negative number to function as one member of an equation. But he refused to admit negative roots. With one exception, all items in the foregoing list may be classified as partial extension by formalism. The extension was incomplete because no free use of negatives was made until the seventeenth century. The extension was formal because it had no basis other than the mechanical application of rules of cal- culation that were known to produce consistent results when applied to positive numbers, and were assumed to be legitimate in the manipulation of negatives. This unbased assumption was to be elevated in the 1830’s to the dignity of a general dogma in the notorious and discredited ‘principle of permanence of form.’ By the middle of the seventeenth century, untrammeled use of negatives had given mathematicians a pragmatic demonstration that the rules of common algebra lead to consistent results. But there was no attempt to go any deeper and put a substratum of postulates under the rickety formalism. The one glimmer of mathematical intelligence in the early history of negatives is the suggestion of Fibonacci that a negative sum of money may be interpreted as a loss. This appears to have been the ff:E&*<ri^p''fd\,fyard the second stage in the evolution of negatives, that of inte'rg-preting the results of formalism in terms of something -which is ■ accepted as consistent. It marks the beginning of two distinct but complementary philosophies of mathematics: the products of mathematical formalism arc to be admitted only if they c: x -.n be put in correspondence with some already established system accepted as self-consistent; all mathematics is a forim-alism without meaning beyond that inv ;ied by the postulates^ defining the formalism. For example, if Euclidean geometry is accepted as self-consistent, and if the formal algebraic operations with complex numbers can be inter- EXTENSIONS OF NUMBER 175 prctcd in terms of that geometry, the formalism of complex numbers is admissible. This is according to the first philosophy, which was that instinctively and subconsciously adopted by Fibonacci in his encounter with negatives. The second philosophy is illustrated by the rules of algebra in any modern elementary text, where a, b, c, • • • , *f-, X, — are displayed, and it is postu- lated that a = a, a + b = b- a, etc. Each philosophy has greatly enriched mathematics. The first, seeking interpretations, may be called synthetic; the second, beginning and ending in a formalism within its own postulated universe, 2 may be termed analytic. The designations arc merely for convenience, and are not intended to recall Kant’s terminology, although the parallel may be suggestive. The development of the number system is the record of a continual interplay between the synthetic and the analytic approaches. For applied mathematics, as in quaternions and the vector algebras that evolved from the geometrical interpretation of ordinary complex numbers, it is the synthetic philosophy that dominates; in pure mathematics the analytic alone is relevant. The history of mathematics holds no greater surprise than the fact that complex numbers were understood, both syn- thetically and analytically, before negative numbers. Accord- ingly, we shall retrace first the principal steps by which complex numbers arrived at mathematical maturity. The negatives will then enter incidentally. From manipulation to interpretation The early history of complex numbers is much like that of negatives, a record of blind manipulations unrelieved by any serious attempt at interpretation or understanding. The first clear recognition of imaginarics was Mahavira’s extremely intelligent remark in the ninth century that, in the nature of things, a negative number has no square root. He had mathe- matical insight enough to leave the matter there, and not to proceed to meaningless manipulations of unintelligible symbols. It is of more than historical interest that Cauchy 3 made the same observation a little less than a tho usan d years later (1847): “{'vc discard] the symbolic sign V— 1, which we repudiate completely, and which we may abandon without regret, because one docs not know what this alleged sign signifies, nor what meaning one should attribute to it.” These sentiments were the 176 THE DEVELOPMENT OF MATHEMATICS origin of Kronecker’s project in 1882-7 for a unified derivation of all extensions of the natural numbers. The next step forward after Mahavira was toward the analytic philosophy of number. Cardan in 1545 regarded imagi- naries as fictitious, but used them formally, as in the resolution of 40 into conjugate complex factors 5 ±y/ —15, without raising any question as to the legitimacy of the formalism. A more vicious species of pure formalism appeared in a totally unwar- ranted conjecture of A. Girard (?1590-?1633, Dutch). Having noticed that some equations of low degrees n have n real roots and that some quadratics have two imaginary roots, Girard inferred that any equation of degree n has n roots, supplying any awkward lack of real roots by guessing that the deficiency would be exactly met by complex roots. Leibniz 4 by 1676 had progressed no farther than Cardan. He gave a formal factorization of x 4 + a 4 , and succeeded in con- vincing himself that he had done something remarkable when he verified by actual substitution that Cardan’s solution of the general cubic in the irreducible case satisfies the equation. He was equally astonished by a similar verification of the expression of a special real radical as a sum of conjugate complexes. The truly astonishing thing historically about Leibniz’ performances with complex numbers is that less than three centuries ago one of the greatest mathematicians in history should have thought that any of these meaningless manipulations were mathematics, or that their outcome was more unexpected than is that of turn- ing a tumbler upside down twice in succession. That a mathe- matician, logician, and philosopher of the caliber of Leibniz could so delude himself, substantiates Gauss’ observation that “the true metaphysics of V - 1” is hard. It also suggests thatmathe- matics really has progressed since the ever-memorable seven- teenth century. In the eighteenth century, blind formalism at last produced a formula of the first magnitude. About 1710, R. Cotes (1682- 1716, English), the man whose death moved Newton to lament, “If Cotes had lived, we might have known something,” stated an equivalent of the result usually called DeMoivre’s theorem in trigonometry. In current notation, i denoting 4 ® V~L Cotes formula 5 is i<f> = log e (cos <j> + i sin <£). DeMoivre’s theorem (1730), cos n<f> + i sin n f> = (cos <f> + i sin 4>) n [= J> f 1 an integer > 0, is an immediate formal consequence. Euler (1743, 1748) extended the last to any n; he also gave the exponents EXTENSIONS Of NUMBER 177 forms of sin d>, cos 4 > — evident from Cotes’ result. Thus, by 1750 trigonometry had become a province of analysis, and all that remained was to derive the analytic formulas with due attention to convergence and to create a self-consistent theory of complex numbers. The first desideratum was met in the third decade of the nineteenth century, by Cauchy; the second, in the last decade of the eighteenth century, by Wessel. So, after about a thou- sand years of meaningless mystery*, the so-called ‘imaginary’ numbers were incorporated into unmystical mathematics. Before proceeding to Wessel and his successors, we recall two noteworthy steps toward what has just been described. John Bernoulli observed the connection between inverse tangents and natural logarithms. This was in Cotes’ direction. Even more significant was the long stride toward a geometrical interpreta- tion of complex numbers taken in 1673 by J. Wallis (1616-1703, English), an original mathematician and at one time a fashion- able preacher. Wallis missed by a hairsbreadth the usual geo- metrical interpretation of complex numbers. But in mathematics a hair may be as thick as a ship’s cable; and Wallis is not usually- credited with Wesscl’s invention. In effect, Wallis represented the complex number x -j- iy by the point (.v, y) in the plane of Cartesian coordinates; what he missed was the use of the y-axis as the axis of imaginarics. It remained for a Norwegian surveyor, C. Wessel 0 (1745- 1818), to take the final step and produce a consistent, useful interpretation of complex numbers in 1797. He modestly called his completely successful effort “an Attempt.” It fully explained what is customarily misnamed the Argand diagram in texts, and mapped the formal algebra of complex numbers on properties of the diagram. J. R. Argand (176S-1S22, French) independently arrived at similar conclusions in 1806. Wcssel’s decisive contribution suffered the misfortune of being published (1799) in a scholarly journal that mathemati- cians were not likely to read. A French translation in 1S97, exactly one hundred years after Wessel had communicated his paper to the Royal Danish Academy, secured its author what- ever reward there may be in posthumous fame. Thus a work that could have hastened the development of the number system might as well never have been written for all the influence it had; and it remained for the great authority of Gauss (1831) to get complex numbers accepted as respectable members of mathematical society. 178 THE DEVELOPMENT OF MATHEMATICS Two possible generalizations were suggested by WessePs interpretation. The geometry of complex numbers was obviously translatable into a description of rotations and dilatations in a plane. Were further extensions of the number system possible by which rotations in space of three dimensions could be described? Or were the complex numbers themselves adequate for the purpose? The answer to the first was to be affirmative, to the second, negative; but this could scarcely have been predicted in 1799 when Wessel’s interpretation was published. The geometrical approach was not the ‘natural’ way to the heart of the problem, although W. R. Hamilton (1805-1865, Irish) was to follow it successfully. But Hamilton, as will be seen, rested content with an algebra adapted to space of three dimen- sions. In mathematics, however, three is no more sacred than any other cardinal number, and the real problem was to extend complex numbers to ‘space’ of n dimensions. The Euclidean program Gauss took as the subject for his doctor’s dissertation (1799) a proof of the fundamental theorem of algebra: an algebraic equation has a root of the form a -f bi, a, b real. (For the precise statement of the theorem, we refer to any text on the theory of equations. The foregoing statement, like others in this account, is intended mereJy-*to recall the theorem.) After Girard’s con- jecture, there^had been attempts at proof, including essays by D’Alembert (1746) and Euler (1749). All were faulty, as were the first and fourth attempts (1799) by Gauss. 7 It may be stated in passing that the fundamental theorem in its classic form, as proved in the theory of functions of a complex variable, is no longer regarded as belonging to algebra. It is supplanted in modern algebra by a statement which is almost a triviality. 8 The basic ideas of the modern treatment go back to Galois (1811- 1832), Dedekind (1831-191^, and Kronecker (1823-1891), not to Gauss. This first serious work of' the greatest mathematician since- Newton convinced him that 'a satisfactory theory of complex numbers had yet to be created. Unaware of Wessel’s work, Gauss himself arrived at a geometrical representation. 9 But the man who gave it as his mature opinion that “Mathematics is t e Queen of the Sciences, and Arithmetic the Queen of Matne- matics,” could not be satisfied with a helpful but irrelevant EXTENSIONS OF NUMBER 179 geometrical picture of tvhat, he believed, was purely a question of number. By 1811, Gauss had convinced himself that a ‘formal’ treatment alone could provide a sound theory of com- plex numbers; and he came within an ace of committing himself to the mysterious principle of permanence which was to guide others to the desired end about a quarter of a century later. But in 1825 he confessed that “the true metaphysics of 'Sj—l” was elusive. By a formal treatment, Gauss meant the deduction of the properties of complex numbers from the accepted postulates of common arithmetic. He sought proofs in the manner of Euclid from definitions and explicit assumptions. We shall return later to the principle of permanence. What Gauss regarded as “the true metaphysics” of complex numbers was invented by him in 1831, six years before Hamilton communicated his independent discovery of the same method to the Royal Irish Academy. The ‘true metaphysics’ banished geometric intuition entirely, defining a -f- bi, a, k real, as the number-couple (a, b) subjected to postulates necessary and sufficient to yield the desired properties of complex numbers as given by algebraic manipulations. I'or example, equality (a, b) (c, d) is defined to mean a — c, b — d\ addition, (a, b) + (c, d), by definition, is {a + c, b T d); multiplication, [a, b) X (c, d) is ( ac — bd , ad + be). The mysterious i has vanished, and the algebra of complex numbers is replaced by what Dc Morgan and others called a “double algebra” of couples of real numbers a,b, c,d, . . . subject only to the accepted laws of arithmetic and common algebra, as a -f- b ~ b + a, ab = ha, a{b 4- c) — ab + ac, etc. The occasion for Gauss’ disclosing his anticipation of Hamil- ton’s method was a letter of 1837 from his old university friend W. Bolyai (1775-1856, Hungarian) in which Bolyai reproached Gauss for having propagated a geometrical theory of complex numbers. Bolyai argued that geometry has no place in the foundations of arithmetic, and that complex numbers should be referred to the real numbers whose arithmetic was assumed to be known. Gr.urs replied that he was of exactly the same epmion and that, in 1831, he had done what Bolyai demanded. He remained of ?hb opinion, and only five year" before his death 180 THE DEVELOPMENT OF MATHEMATICS emphasized that the “abstract,” postulational method is the desirable approach to complex numbers. This method is now fairly common in texts on college algebra. Anyone seeing the algebra or arithmetic of number-couples for the first time might be pardoned for thinking it a sly subter- fuge, or at best a beating of the imaginary devil round a sup- posedly real bush. Familiarity corrects misapprehension; and when the suggestive notation ( a , b ) for couples is extended to triples ( a , b, c ), and beyond, to sets of n ordered real numbers or elements of a field, with appropriately defined laws of addition and multiplication, the creative power of Hamilton’s simple invention becomes evident. Multiple algebra, with its innumer- able applications to the sciences, was already in sight when Hamilton replaced a + hi by ( a , b). He himself elaborated the algebra and geometry of number-quadruples ( a , b , c, d) in his quaternions; Grassmann almost simultaneously took a more general point of view and created the algebra of number-?z-ples, (ai, a 2 , , a n ). We shall resume this in a later chapter; for the moment, we follow the consequences of the resumption of the Euclidean methodology by Gauss and Hamilton. After about twenty-three centuries of sightless wandering, arithmeticians and algebraists opened their eyes and saw what Euclid had done: definitions, postulates, deduction, theorems. They then took a long stride ahead. It may have been clear to Euclid that his geometry was that of a postulated, ideal universe having no necessary connection with an intuitively perceived ‘real world’ of common experience; but if so, he did not convey the full import of his philosophy to his successors. Euclid’s geometry, or any other mathematical system constructed on the deductive pattern, is now almost universally regarded as a free and arbitrary creation of the mathematician constructing the system, whether the initial impulse came from experiences of the material world sublimated into abstractions, or whether it originated in formal extensions of algebraic symbolism, as in the passage from number-couples to ordered sets of n real numbers. The philosophy behind the Euclidean program as now conceived is analytic. It seems singularly appropriate that the conception of alge- bra as pure formalism should have first appeared in that country which above all others has revered Euclid. It was an Englishman, G. Peacock (1791-1858), at one time Lowndean professor in Cambridge University, later a Dean of Ely, who first 10 (18 , EXTENSIONS Or NUMBER 3S1 1845) perceived common algebra 11 as an abstract hypothctico- dcductivc science of the Euclidean pattern. Peacock was not an ‘important’ mathematician in the ac- cepted sense of wide reputation; so possibly the following is a just estimate of his place in mathematics: “He was one of the prime movers in all mathematical reforms in England during the first half of the 19th century, although contributing no original work of particular value.” 12 He was merely one of the first to revolutionize the whole conception of algebra and general arithmetic. The Euclidean program advocated by Peacock was developed by the British school, notably by D. F. Gregory (1813-1844, Scotch), and A. Dc Morgan; but it did not become widely known until H. Hankcl (1839-1873) in 1867 expounded it with insight and massive German thoroughness. Hankcl also reformulated the principle of permanence of formal operations, which had been stated in less comprehensive terms by Peacock: “Equal expressions couched in the general terms of universal arithmetic arc to remain equal if the letters cease to denote simple ‘quan- tities,’ and hence also if the interpretation of the operations is altered.” For example, ah — ba is to remain valid when a, b are complex. It is difficult to see what the principle means, or what possible value it could have even as a heuristic guide. If taken at what appears to be its face value, it would seem to forbid ah — —ba, one of the most suggestive breaches of elemental-}- mathematical etiquette ever imagined, as every student of physics knows from his vector analysis. As a parting tribute to the discredited prin- ciple of permanence, we note that since 2 X 3 = 3 X 2,itfol!ows at once from the principle that \^2 X V3 — X \5. But the necessity for proving such simple statements as the last was one of the spurs that induced Dedekind in the 1870’s to create his theory of the real number system. According to that peerless extender of the natural numbers, “Whatever is provable, should not be believed in science without proof.” 13 The device of number-couples, invented to exorcise the imaginary and reduce the theory of complex numbers to that of pairs of real numbers, also banished rational fractions and negative numbers. Thus, for negatives, — is replaced by [«:, ir, -4 «], where v: is an arbitrary positive number; zero is {«:, and r. is -f n, »:]. As the details are available in stand- ard texts, we pass on. The last and most difficult step in reducing 182 THE DEVELOPMENT OF MATHEMATICS all ‘number’ to the natural numbers 1, 2, 3, . . . concerned the real irrationals. This arithmetized analysis. Between the final step and the formalizing of algebra and arithmetic by Peacock, De Morgan, Hamilton, and others, the natural numbers were vastly extended in another direction, that of algebraic numbers, beginning with Gauss in 1831 and continu- ing into the twentieth century. Concurrently, the generalizations of number-couples to multiple algebra were developed. Another type of arithmetization, originating in 1801 in the work of Gauss and reaching one of its climaxes in Kronecker’s work of 1882-7, incidentally provided another means of reducing all numbers to the natural numbers. This will be described in the following chapter. The Euclidean program, which would ultimately reduce all mathematics to a pure formalism, had its opponents as well as its partisans in the nineteenth century as it has had since. To illustrate the ironies of prophecy, we recall the vigorous attack delivered in 1882 by a distinguished analyst, P. du Bois-Rey- mond (1831-1889, German), whose penetrating researches contributed much to the progress of analysis in its second heroic age — that of Newton and Leibniz being the first — in the nine- teenth century. The program of formalism, du Bois-Reymond declared with considerable passion, would replace mathematics by “a mere play with symbols, in which arbitrary meanings would be attached to the signs as if they were the pieces on a chessboard or playing cards.” He went on to prophesy that such a ‘meaningless’ outcome would fritter out in barren efforts and be the death of mathematics as Gauss had pictured the Queen of the Sciences. Since 1920, mathematics for one highly produc- tive school has become exactly what the prophet feared it might. Those who call themselves formalists revel in their endless game of chess and exult that it has no meaning whatever beyond the rules of the game. Ultimate realities and eternal truths, at least in mathematics and science, suffered an eclipse in the twentieth century. Thus has ended one quest after the meaning of number; and this conclusion, disconcerting to some, was reached by following the same road that Euclid took. It was D. Hilbert’s (1862-1943, German) close scrutiny of the postulates of elementary geometry, in an endeavor to put a solid foundation under that venerable i somewhat palsied science, that led him to a similar inspection o the bases of common arithmetic. Addressing the second inter- EX TENSIONS OF NUMBER 1S3 national congress of mathematicians in 1900, Hilbert observed 15 that the noncontradiction of the postulates of geometry is dem- onstrated by constructing a suitable domain of numbers such that, to the geometrical postulates, there correspond analogous relations between the numbers of the domain. Consequently, any contradiction in the conclusions drawn from the geometrical postulates would necessarily be recognizable in the arithmetic of the domain. Thus the self-consistency (noncontradiction) of the postulates of geometry is referred to the self-consistency of the postulates of arithmetic. Consequently Hilbert emphasized as one of the outstanding unsolved problems of mathematics in 1900 the proof that, by proceeding from the postulates of arith- metic, it is impossible to reach contradictory results by means of a finite number of logical deductions. The problem was still open in 19-15. Attempts to solve this seemingly elementary problem were in part responsible for the formalistic — chess-playing — school of mathematical philosophy led by Hilbert, the foremost mathematician of his generation. Enough has been said to indicate the fundamental impor- tance of the natural numbers for all mathematics, not merely for arithmetic and its algebraic extensions, and to suggest that Euclid’s methodology is as vital in modern mathematics as it was in ancient. It will be well before proceeding to further extensions of number to cast up the account of Pythagoras with mathematics thus far. Pythagoras to 1900 Glancing back over the confused effort to incorporate imagi- naries with the reals in one self-consistent number system, we note the curious fluctuations in mathematical creed accompany- ing the struggle. The abrupt check experienced by the Pytha- goreans in their encounter with irrationals practically abolished mensuration in orthodox Greek mathematics, and the investiga- tion of number independently of its geometrical representation all but ceased, Euclid’s summary* of arithmetica being a partial exception. The academic Greek mathematicians were at ease with number only when it was geometrized into ‘magnitude’ — a vague concept whose tenabilitv they seem never to have questioned. Thus number was supposed to be apprehensible through form, the opposite of what the Pythagoreans first held and of what a majority of mathematicians have believed since Descartes. 184 THE DEVELOPMENT OF MATHEMATICS The diagrams of Wessel and Argand were an unseasonable reversion to pre-Cartesian mathematics. Exactly nothing is provable by the geometrical representation of complex numbers unless it be assumed that the underlying geometry is founded consistently. Gauss also, we have seen, at first sought to ‘justify’ imaginaries geometrically, but later decided that this was a mistake. In all these earlier developments, geometry was ac- cepted without question as the irreversible court of last appeal. But with increasing sophistication, it was perceived that the geometrical justification is merely disguised arithmetic, real numbers entering with the coordinates of a point in the plane of complex numbers. The geometrical interpretation is thus left without a foundation until the real number system is firmly based in self-consistency. Going deeper, Hilbert in 1899 resumed the Pythagorean program for all geometry, referring form to number and demanding a proof of noncontradiction for the real number system, or even for its subset of rational integers. Hamilton also was a Pythagorean in his escape from geom- etry into number-couples. His was the more suggestive method for future extensions to hypercomplex numbers. But he was less critical than Hilbert, in that he took the self-consistency of the real number system for granted. The modern attack, as in abstract algebra, attempts to strip Hamilton’s number-couples ( a , b ) of all arithmetical connota- tions by postulating that the ‘coordinates’ a, b are defined by the postulates of an abstract field. The last vestige of number as the Pythagoreans conceived it has been sublimated from the ‘mean- ingless marks’ a, b and their equally ‘meaningless rules of com- bination.’ But the problem of proving that the rules will never produce a contradiction is not eliminated by manipulating a set of postulates which, by assumption, completely define the mathematical system deducible from them. In escaping from form to number, and back from number to form, thence again to number, and finally into complete ab- straction, mathematicians from Pythagoras to Hilbert have sought to validate their creations by deductive reasoning. Hilbert was the first to recognize the futility of such vacillation until deductive reasoning as applied in all mathematics should itself be shown to be incapable of producing contradictions. This put the whole Pythagorean program on trial; for the cardina hypothesis of the Pythagoreans assumed that number and form may be consistently described by deductive reasoning. The fina EXTENSIONS OF NUMBER 185 question, then, is, How far can mathematical deduction be trusted if it is not to produce contradictions such as l X is equal to B, and A is not equal to B' ? This is debated in the symbolic language of reasoning foreseen by Leibniz. Some of the conclu- sions will be noted in the last chapter. For the moment it is sufficient to observe that the ability of modern mathematics to discuss such questions profitably is one of its titles to superiority in power over its predecessors. And, whatever the outcome is to be, the admitted utility' of complex numbers, whether conceived as affixes of points in a plane or as number-couples, in both pure and applied mathe- matics, will doubtless remain substantially unaffected. CHAPTER 9 Toward Mathematical Structure 18014910 Three new approaches to number, in 1801 and in the 1830’s, were to hint at the general concept of mathematical structure and reveal unsuspected horizons in the whole of mathematics. That* of 1801 was the concept of congruence, introduced by Gauss in what many consider his masterpiece, the Disjuisitiones arith- meticae, published when its author was twenty-four. To this and the revolutionary work (1830-2) of E. Galois (1811-1832, French) in the theory of algebraic equations can be traced the partial execution of L. Kronecker’s (1823-1891, German) revolu- tionary program in the 1880’s for basing all mathematics on the natural numbers. The same sources are one origin of the modern abstract development of algebraic and geometric theories, in which the structure of mathematical systems 1 is the subject of investiga- tion, and it is sought to obtain the interrelations of the mathe- matical objects concerned with a minimum of calculation. ‘Structure’ may be thought of for the present in any of its intui- tive meanings; it was precisely defined in 1910 by the mathematical logicians. It might be compared to morphology and comparative anatomy. We shall approach mathematical structure through the union effected in the nineteenth century between algebra and arithmetic. Abstraction and the recent period From the standpoint of mathematics as a whole, the method- ology of deliberate generalization and abstraction, culminating in the twentieth century in a rapidly growing mathematics o 186 TOWARD MATHEMATICAL STRUCTURE 1S7 structure, is doubtless the most significant contribution of all the successive attempts to extend the number concept. But at ever}' stage of the progression from the natural numbers 1, 2, 3, ... to other types of numbers, each of several fields of mathematics adjacent to arithmetic v.-as broadened and enriched. New acquisitions in other fields reacted reciprocally on arithmetic. For example, the first satisfactory theory of ordinary complex numbers to become widely known was that of Gauss (1831), devised to provide a concise solution for a special problem in diophantinc analysis: If p, q arc primes, what conditions must p, q satisfy in order that at least one of the equations ari = qy + p, z* = pee + shall be solvable in integers x, y, z, r r? The theory of complex numbers necessitated a radical revi- sion and generalization of the concept of arithmetical divisibility, which in turn suggested a reformulation of certain parts (inter- sections of varieties) of algebraic geometry. The latter in its turn was partly responsible for further generalizations (modular systems) in the algebraic arithmetic — or arithmetical algebra — of the twentieth century. The like may be observed in the creation of the numerous vector algebras invented during and after the 1840's for applica- tion to the physical sciences. The first of these evolved directly from the vectorial interpretation of ordinary complex numbers. The extension in the lS40’s of vector algebra in a plane to space of more than two dimensions was one origin of the hypercomplex number systems of algebra, and these again supplied arithmetic with new species of integers. The development of the correspond- ing arithmetic in turn reacted, particularly in the twentieth century, on the algebra in which it had originated. It would seem to be incorrect, therefore, to say that any one division of mathematics was alone responsible for the steady progression since 1800 from the special and detailed to the abstract and general. The forward movement was universal, and each major advance in one department induced progress in others. In following this development, one misapprehension above ail others that might be possible is to be particularly guarded against. Those who are not mathematicians by trade arc some- times inclined to confuse generality with vagueness, and abstrac- tion with emptiness. The exact opposite is the ease in the mathematical generalizations and abstractions with which we shall be concerned here. Each, on appropriate and definitely prescribed specialization, yielded the specific instances from 188 THE DEVELOPMENT OF MATHEMATICS which it had evolved. The theory of hypercomplex numbers for example, contains as a mere detail that of ordinary complex numbers; and once the general theory of hypercomplex number systems has been elaborated, the special theory of ordinary complex numbers follows automatically. Moreover, each gen- eralization gives in addition a whole universe of mathematical facts distinct from those in the special instances from which the generalization proceeded. It was remarked in the Prospectus that the separation of all mathematical history into a remote period, to 1637, a middle 1638-1801, and a recent, 1801 — , distinguishes three well-marked epochs in the development of mathematics. Following the rapid growth of arithmetic and algebra, we shall see that in the passage from the middle period to the recent there was a profound change in the quality of mathematical thought and its objectives. This change is most simply observed, perhaps, in the evolution of the number concept. It is the item of greatest interest to be noticed in this chapter and the following. Geometry might have been considered instead of arithmetic to exhibit the same change. But as the transformed algebra and arithmetic played important parts in the expansion of geometry, it seems more natural to consider them first. It is to be borne in mind, however, that while arithmetic and algebra were being transformed into shapes the mathematicians of the eighteenth century would not have recognized as mathematics, geometry and analysis were under- going corresponding transformations. Prospect The abstract approach of the 1830’s to algebra parallels the epochal advance in geometry made simultaneously with the publication in 1829 of N. I. Lobachewsky’s (1793-1856, Russian) non-Euclidean geometry. This also stems from 1800 or earlier in the preparatory work of Gauss and others. As this properly belongs to geometry, it will be discussed in that connection. The relevant detail here is that geometers and algebraists perceived almost simultaneously that mathematical systems are not supernaturally imposed on human beings from without, but are free creations of imaginative mathematicians. Lobachewskys new geometry was the earliest mathematical system to be recognized as such a free creation. It provided the first proof of the complete independence of a particular postulate (Euclid s postulate of parallels) in a system which tradition and common TOWARD MATHEMATICAL STRUCTURE 389 sense had agreed must contain that postulate. The significance of this radical step in methodology was only slowly appreciated; and it would seem that the almost simultaneous advance of algebra and arithmetic in a parallel direction was more directly responsible than geometry for the modern abstract view of mathematics. The explicit recognition, by the British school in the 1830’s, of common algebra as a purely formal mathematical system shortly led to a revolution in arithmetic and algebra of signifi- cance comparable to that precipitated by non-EucIidean geometry'. Hamilton’s rejection (1843) of the commutative ‘law’ (postu- late) of multiplication, in his invention of quaternions, opened the gates to a flood of algebras, in which one after another of the supposedly immutable ‘laws’ of rational arithmetic and common algebra was either modified or discarded outright as too restric- tive. By 1850 it was clear to a majority of creative mathemati- cians that none of the postulates of common algebra, which up to 1843 had been thought necessary for the self-consistency of symbolic reasoning, was any more a necessity for a noncontradic- tory algebra than is Euclid’s parallel postulate for a self-con- sistent elementary geometry. To the astonishment of some, it was found that the modified algebras, such as Hamilton’s quaternions, were adaptable to mechanics, geometry, and mathe- matical physics. The dead hand of authoritative tradition had been brushed aside; mathematics was free. As G. Cantor (1845- 1918, Germany), one of the boldest extenders of the number concept, was to say about three-quarters of a century' later, “the essence of mathematics is its freedom.” No mathematician, not even Gauss, could have conceived such a thought in 1801. The accomplished facts of the revolutions in geometry and algebra of the 1830’s and lS40’s made freedom conceivable. From supernaluralism to naturalism The change during the nineteenth century from what may be called Platonic supcrnaturalism to modern naturalism in mathe- matics is reflected in three aphorisms, the first of which may' have expressed the Greek reverence for synthetic geometry; the second, the early ninctccnth-ccntury worship of arithmetic and nnalv sis; and the third, the final admission that mathematics is made by men. The second and third, following the example of the first, were phrased in classic Greek. 190 THE DEVELOPMENT OF MATHEMATICS Plato is said to have asserted that “God ever geometries’’- C. G. J. Jacobi (1804-1851, German), a great arithmetician and analyst, declared that “God ever arithmetizes”; while J. W. R. Dedekind (1831-1916, German), first and last an arithmetician wrote as the motto for his famous essay on the nature of number {Was sind und was sollen die Zahlen?, 1888), “Man ever arithmetizes.” A proposition less open to objection than the last today would be “Man attempted to arithmetize during the second half of the nineteenth century, and came to grief so doing early in the twentieth.” Although this is less elegant than the original, it is closer to historical fact. Nevertheless, man’s failure as yet to com- plete the Pythagorean program of arithmetizing mathematics and the universe has been, and is, a potent stimulant to the con- tinual creation of new and interesting or useful mathematics. During the nineteenth century the physical sciences were given to solving the universe by dissolving it in vast generaliza- tions distilled from inadequate data. So powerful were some of these solvents that they dissolved themselves. Having learned by disconcerting experience that the universe is not to be solved between breakfast and lunch, the physical sciences took a more modest view of their function, and in the early twentieth century, after critical introspection, contented themselves with consistent descriptions intelligible to instructed human beings. Universe- solving went out of fashion temporarily about the time of the first world war. Mathematics in the meantime was experiencing similar diffi- culties in its abortive struggle to comprehend its own vast empire in the all-inclusive generalization of Pythagoras. What is to be the outcome is not yet predictable. But two things may be reasonably conjectured. The Pythagorean project of deriving all mathematics from number will continue for many years to suggest new accessions to mathematics, and it will remain essentially as the Pythag- oreans imagitfpd it in their unmystical moments. If form, for example, is to be better described by our successors in terms of something other than number, we have no inkling at present of what that something may be, unless it be symbolic logic or analysis situs, themselves in partial process of arithmetization. We may also conjecture that mathematics, like the physica sciences, will take a less inflated view of itself as a result of its critical self-analysis. There will be less mysticism in the mathe- TOWARD MATHEMATICAL STRUCTURE 191 matics of the future than there has been in that of the past, and fewer grandiose claims to immortality and eternal truth. Mathe- matics will become less self-conscious, less introspectively critical, and more boldly creative. It will resign its soul to the metaphysicians for such tortures as they may choose to inflict, feeling nothing; for it will continue to serve with its living body the purposes of the men who create it to meet human needs rather than to be the plaything of sterile philosophies. The very implements of torture — symbolic logic, for example — were devised in mathematics itself as by-products of more immedi- ately useful inventions. It is the latter that are of vital significance in a scientific civilization; the by-products too often carry with them the cold smell of a mildewed scholasticism. The spirit of the Middle Ages, which the successors of Galileo and Newton imagined they had laid forever in science, stirs again in the twentieth- century disputes concerning the nature and meaning of number. Ignoring these for the present, we shall continue with the more profitable arithmetic of 1801 which, a century later, was to deliquesce in metaphysics. But history will compel us to return to these disputes. Our attitude in what follows will be that of Molierc’s despised ‘‘average sensual man,” who seeks through science merely to make life less barbarous for himself and his fellows, and who is content to leave what professed humanists call “the really important questions” to God and the philosophers. Congruence from 1801 to 1887 ‘Congruence,’ like ‘analysis,’ ‘formal,’ ‘ideal,’ ‘functional,’ ‘analytic,’ ‘normal,’ ‘conjugate,’ ‘modulus,’ ‘integral,’ and a dozen others, is one of those overworked technical terms in mathematics which appear to have been invented to confuse the uninitiated by a multitude of meanings having no connection with one another. Congruences in higher geometry, as in con- gruences of lines or circles, arc unrelated to congruence in elementary geometry, as in congruent triangles; and congruence in arithmetic, with which we arc concerned here, has nothing in common with congruence elsewhere. Nor arc the ideal elements of projective geometry significantly connected with ideals as in arithmetic and algebra. Gauss in 1S01 defined two rational integers a , b to be con- gruent with respect to the rational integer modulus m if, and only 192 THE DEVELOPMENT OF MATHEMATICS if, a , b leave the same remainder on division by m; and he expressed this by writing a s= b mod m. Otherwise stated if a 2 = b mod m, then a — b (or b — a) is a multiple of m, and conversely; and x s 0 mod m asserts that # is exactly divisible by m. This simple but profound invention is one of the finest illus- trations of Laplace’s remark that a well-devised notation is sometimes half the battle in mathematics. Writing ‘x is divisible by ml as x = 0 mod m at once suggested to Gauss extremely fruitful analogies between algebraic equations and arithmetical divisibility. The last is one of the central and most elusive con- cepts of all arithmetic. It is not this technical aspect of con- gruence, however, that is of primary importance for our immediate purpose, but another, of far deeper significance, which was perceived only by the successors of Gauss. If Gauss did foresee this, he seems to have left no record of the fact. To bring out the point, of the first importance for an under- standing of modern mathematical thought, we must return for a moment to a hypothetical prehistory, long before Pythagoras and even prior to Sargon. Abstraction, to judge by the behavior of contemporary primitives, is not by any means ‘natural’ to Rousseau’s carefree savage. Numbers first were nouns as con- crete as father and mother, an early instance, perhaps, of ‘one,’ ‘two.’ No trace survives of the actual passage from concreteness to abstractness, when ‘two’ was realized as applicable to a couple of parents, a stick and a stone, or any other of its innumerable manifestations; and we can only imagine the dismay of human beings when they were first overwhelmed by the appalling revelation that the natural numbers have no end. Traces of the attempt to cope with that first deluge of knowledge survive in the symbolism, meaningless to us, of number mysticism. Sym- pathetically viewed, all that prehistoric nonsense was the outcome of men’s first groping efforts to regiment the generative freedom, c n into n + 1,’ of the numbers in the unending sequence 1, 2, 3, + 1, .... If only some finite restraint could dominate the endless generations of numbers, they would be less terrifying. It must have given the mind that first perceived that odd and ‘even’ suffice to comprehend all the natural numbers a sense of almost supernatural power. The endless sequence after all was no more mysterious than humanity itself, which could be subsumed under ‘male’ and ‘female.’ Accordingly, the mat e- TOWARD MATHEMATICAL STRUCTURE 193 matically useful separation of the natural numbers into only two classes was made more concretely satisfying to the primitive mind by calling odd numbers male and even numbers female. Arithmetic and numerology thereafter flourished together in happy and fruitful symbiosis. But however nonsensical the numcrological fruits of that early union may now appear, its occasion was the urge to comprehend an infinite totality in finite terms, and hence to bring the infinite within the grasp of a finite syntax. Gaussian congruence has proved the most fruitful of all classifications of the rational integers 0, +1, ±2, ±3, . . . into a finite number of classes, as may be appreciated on inspecting any elementary text on the theory of numbers. What Gauss could not have foreseen was that his invention of mapping one assemblage, finite or infinite, of individuals on another, by classifying the individuals in the first set with respect to some relation having the abstract properties of reflexiveness, sym- metry, and transitivity, shared by his relation of congruence, was to become a guiding principle to the structure of algebraic theories. With the gradual evolution of this and similar ideas, mathematics transcended the Pythagorean dream and, as in the theories of groups, fields, point sets, symbolic logic, etc., escaped from the natural numbers into a domain where number is irrelevant and the structure of relations is the subject of investigation. The concepts mentioned above being fundamental in modern mathematics, we shall recall their definitions. A relation, denoted by is said to be binary with respect to the members a, b, c, ... of a given class of things (which need not be ‘numbers’ of any kind) if a ~ b is either true or false for any a, b in the class. If a ~a for every a in the class, ~ is reflexive; if a ~ b implies b ~ a, ~ is symmetric; and finally, if a ~ b and b ~ c together imply a ~ c, ~ is transitive. A relation such as ~ is called an ‘equivalence relation’ for the given class. Equality, =, is a simple instance of If in, a,b,c, . . . arc rational integers, and in pi 0, and if a ~ b is interpreted as a ~ b mod in, it is easily verified that this Gaussian congruence is an equivalence relation. Further, congruence is preserved under addition and multiplication: if .v e= a mod in, y — b mod m, then .v -f- y -- a + b mod in, and .yy ab mod m. Any equivalence relation separates its class, whether finite or infinite, into subclasses, all those members, and only those, 194 THE DEVELOPMENT OF MATHEMATICS of the whole class that are equivalent to a particular member (and hence by transitivity to one another) being included in a particular subclass. Any member of a subclass may be taken as representing the entire subclass. Congruence with respect to the positive integer modulus m separates all the rational integers into precisely m classes, whose representatives may be taken as 0, 1, 2, . . . , m — 1. Congruence is a typical example, and historically the first, of the modern methodology of mapping an infinite totality on a comprehensible finite set. The theory of arithmetical congruence as developed by Gauss and his succes- sors belongs to the higher arithmetic, and will be noted in that connection. Our present interest in congruence is in yet another direction, of deeper significance than the technical applications to arithmetic for mathematical thought as a whole. In the preceding chapte r we noted Cauchy’s objections (1847) to the symbol i ( = V~l)- An immediate extension of Gaussian congruence to congruences between polynomials in one variable (more properly, 'indeterminate’) x provided Cauchy with the escape into the illusory ‘reality’ he so ardently desired. If F, = jf) A r x m ~ r , and M, - ^ B s x n ~\ r «= 0 «"0 are polynomials, with in ^ n and A 0 B 0 ^ 0, there is exactly one polynomial R of degree n — 1, and exactly one polynomial Q, such that Bo'~ n+i F = QM + R. Cauchy wrote the particular case of this in which B 0 = 1 as a congruence, F = R mod M, and imitated the Gaussian theory in his easy development of such congruences for polynomials. For the particular modulus x 2 + 1( = M), Cauchy found that his ‘residues’ R had all the formal properties of complex numbers, his ‘x’ taking the place of ‘t.’ He was thus enabled to construct a wholly ‘real’ algebra abstractly identical with (having the same structure as) that of complex numbers. A moment’s reflection will show why his ingenious device succeeded. It offered an alternative to Hamilton’s number-couples. It seems rather surprising that Cauchy, having gone so far, should not have continued to the expulsion of negatives from his paradise of ‘real,’ ‘existent’ numbers, for they surely are as ‘unreal’ and as ‘inexistent’ as i to the Pythagorean mind. Natu- rally enough, the Cauchy who in 1821 had given the first satis- factory definitions of limits and continuity in the calcu us TOWARD MATHEMATICAL STRUCTURE 195 noticed nothing demanding reform in 1847 in the continuum of real numbers with their non-denumcrabie infinity of irrationals. A thoroughgoing Pythagorean would have expelled the real irrationals along with i. Cauchy extended his invention to what he called algebraic keys; but, as these were very special instances 2 of some of the algebras 5 already implicit in the work (1S44) of H. G. Grass- mann (1809-1877, German), they have rather missed fire. The prolific Cauchy passed on to new creations more conformable to his passion for analysis. His ingenious suggestion went unnoticed for forty years, when (1S87) it reappeared, greatly amplified, in the arithmetical program of Kronecker.' 1 Here at last was the modern Pythagoras. Gauss is said to have ascribed an ‘external reality 5 to ‘space 5 and ‘time, 5 while reserving for number the ideal purity of a ‘creation of the mind.’ Kronecker denied this philosophy, insisting that geometry and mechanics are expressible wholly in terms of relations between numbers, and by numbers he meant the positive integers 1, 2, 3, . . . . Thus, for him, the continuities ‘space 5 and ‘time, 5 fused in the concepts of kinematics, had meaning only in terms of the ineradicable discontinuities of these same God-given natural numbers. Continuity had no meaning; all was discrete. To show how his subversive program might be carried out, Kronecker expelled negative numbers by means of congruences to the modulus j + 1, precisely as Cauchy had banished imagi- narics a -f- hi with his modulus f 2 + 1. Since only the natural numbers existed for Kronecker, he exorcised rational fractions by a similar magic, introducing (in effect) a new symbol, or ‘indeterminate, 5 for each objectionable fraction. To dissipate -J, for example, it sufficed to use congru ence s to the compound modulus •}/; -f- 3 j. An irrational, say could be dispensed with by a new indeterminate t and an additional modulus l- -j- 2. Arithmetic, algebra, and analysis began to grow somewhat complicated. But that was beside the point. In the work cited and in an extensive earlier memoir (1882), Kronecker outlined in some detail how the program of Pythag- oras could be realized in modern mathematics. Whether such a project was worth doing is irrelevant. Kronecker was interested primarily in showing that the Pythagorean vision could be materialized. Provided that were once demonstrated, careless mortals presumably were to be permitted to use negatives and irrationals in the customary manner and in the usual notations, 19 6 THE DEVELOPMENT OP MATHEMATICS on the understanding, however, that they admitted their work- able mathematics to be merely a convenient shorthand for the only true mathematics, that of Kronecker’s modular systems. It would be interesting to know what Gauss would have thought of this devastating outcome of his simple device of writing ‘n is divisible by m’ as n — 0 mod m. Eminent mathe- maticians have called it anything from anarchy to hocus-pocus. Kronecker, however, might have recompensed himself at the expense of the nineteenth-century analysts, had he lived to participate in the debates of the twentieth century on the con- sistency of classical analysis. For few in the 1940’s would have written with the resolute conservatism of E. W. Hobson (1856— 1933, English) in 1921: Kronecker’s ideal . . . that every theorem in analysis shall be stated as a relation between positive integral numbers only, . . . , if it were possible to attain it, would amount to a reversal of the actual historical course which the science has pursued; for all actual progress has depended upon successive generalizations of the notion of number, although these generalizations are now regarded as ultimately dependent on the whole number for their founda- tion. The abandonment of the inestimable advantages of the formal use in Analysis of the extensions of the notion of number could only be characterized as a species of Mathematical Nihilism. Apart from its Pythagoreanism, Kronecker’s effort left a useful residue, his theory of modular systems. This provides an alternative approach to algebraic numbers, Dedekind’s being that usually followed. One of the elementary by-products of Kronecker’s algebra (1882) provided a rigorous theory of elimination for systems of polynomials in any number of variables. This rendered obsolete many unsatisfactory attempts, particularly by algebraic geome- ters, to give sound proofs for the speciously simple formalism of such methods as grow out of J. J. Sylvester’s (1814—1897, English) of 1840, and E. Bezout’s (1730-1783, French) of 1764. The latter was also invented independently by L. Euler (1707- 1783, Swiss). The usual textbook discussion is still in the spirit of 1764, although there are honorable exceptions. The same cynical fate awaited Kronecker’s reduction of all mathematics to the natural numbers that seems sooner or later to nullify all human attempts to solve the universe at one stroke. It does not appear to have occurred to him that the natural numbers themselves might some day be put on trial as he had tried all other numbers and found them wanting in meaning. TOWARD MATHEMATICAL STRUCTURE 197 Any savage might have suggested such a possibility; but it remained for the mathematical logicians of the early twentieth century to demonstrate the possibility up to the hilt and beyond. A -period of transition We must briefiv indicate the involuntary participation of E. Galois (1S11-1S32, French) and N. H. Abel (1S02-1S29, Norwegian) in the development of Kronecker' s Pythagoreanism. Galois himself adhered to no such creed. Nor did Abel. But it was in the attempt to understand and elucidate the Galois theory of equations, left (1S32) by its young author in a rather frag- mentary and unapproachable condition, that Kronecker ac- quired some of his skill. Both Kronecker and Dedekind, two of the founders (E. E. Rummer [1810-1893, German] being a third) of the theory of algebraic numbers, were inspired partly by their scrutiny of the Galois theory to begin their own revolu- tionary work in algebra and arithmetic. Kronecker also began some of his researches in the arithmetization of algebra with a profound study of abelian equations. Galois and Abel mark the beginning of one modern approach to algebra. The transition from highly finished individual theorems to abstract and widely inclusive theories is plainly evident in the algebra of Gauss contrasted with that of Abel and Galois. The like is seen in other fields, as will appear presently. This transition took place about 1S30, contemporaneously with the abstract approach of the British algebraists. Younger than Gauss by thirty-four years, and dying twenty- three years before him, Galois now, curiously enough, seems more modern than Gauss. A single example will suffice to sub- stantiate the radical distinction between the two minds. The occasion for Gauss’ making mathematics his lifework was his spectacular discovert- at the ace of nineteen concerning the construction of regular polygons by means of straightedge and compass alone. Gauss proved that such a construction is possible if, and only if, the polygon has ?; sides where r. is an integer of the form 2* Pjpr s *5 0, in which pi. p*. . . . , p. arc r different primes, each of the form a power of 2 plus 1. The alge- braic equivalent of this theorem, concerning binomial equations, is partly developed in the seventh and last section of the Dis- quisitir.r.es arithmeticae. This particular work marks the end of its era in mathematical outlook. TOWARD MATHEMATICAL STRUCTURE 199 These two liberators of algebra are among the nineteenth century’s major mathematical prophets. Both were richly gifted in many things besides mathematics. Hamilton at the age of thirteen was an accomplished classicist and a proficient linguist in the oriental as well as the European tongues; Grassmann was a profound scholar of Sanskrit. At twenty-seven Hamilton was famous as the result of his mathematical prediction of con- ical refraction, a deduction from his comprehensive theory of systems of rays in optics; and by thirty he had practically com- pleted his fundamental work in dynamics, an advance beyond Lagrange comparable to Lagrange’s beyond Euler. At the age of thirty-eight (1843) he overcame the difficulty which had pre- vented him from extending the algebra of coplanar vectors to a theory of vectors and rotations in space of three dimensons. Ide discovered that the commutative law of multiplication is not necessary for a self-consistent algebra. Thereafter Hamilton’s scientific life was devoted to the elaboration of his theory of quaternions, in the mistaken hope that the new algebra would prove the most useful addition to mathematics after the dif- ferential and integral calculus. Plonors were showered on Hamilton; none fell on the less fortunate Grassmann. Neither had a particularly happy life. Hamilton was afflicted by domestic troubles and personal weak- nesses; Grassmann supported himself, his wife, and nine children by elementary teaching, a profession for which he was eminently unsuited. A steadfastly pious man, Grassman trusted that if his contemporaries failed to reward his signal merits, the Lord would. He never complained of the torments he endured from the young savages he was meagcrJy paid to civilize. His avocations were his life — the Sanskrit classics, philosophy, phonetics, harmony, philology, physics, theology, politics being among the extraordinary miscellany. But with the possible exception of theology, Grassmann’s creation of “a new branch of mathe- matics in 1840-4 gave him the most abiding satisfaction. Here his inventive imagination and his perverse originality had free play. His theory of extension ( Ausdehr.ungslchre ), in which PInmilton’s quaternions are a potential detail, was first published in 1844, about a year after Hamilton had found the clue to his problem of rotations in the equations = i 1 ~ h- — ijk — — 1 defining the quaternion units :, /, It has often been observed that it is not healthful for a mathematician to be a philosopher. Whether or not this is a general theorem, it was certainly true in the unfortunate Grass- 200 THE DEVELOPMENT OF MATHEMATICS mann’s case. Endowing his theory with the utmost generality it could support, he all but smothered it in philosophical abstrac- tions. This was one of the greater tragedies of mathematics Gauss looked the Ausdehnungslehre over, and blessed it with his qualified approval. It was partly in the same directon, he said as he himself had taken almost half a century earlier. But it was too philosophical with its “peculiar terminology” even for Gauss, himself no inconsiderable amateur of philosophy. Gauss in the meantime had recorded his own independent discovery of Hamilton’s quaternions. In a brief abstract which he never published, 6 ascribed to the year 1819, Gauss wrote out the fundamental equations of what he called mutations in space, essentially quaternions. Grassmann continued his efforts to gain recognition for his own incomparably more general theory. Eighteen years (1862) after the first publication of his book, he brought out a com- pletely revised, greatly amplified, and somewhat less incom- prehensible version. 5 But a mathematician who has once been seriously called a philosopher might as well have been hanged for all the hearing he is likely to get from his fellow technicians. The second edition followed the first into temporary oblivion, Grassmann abandoned mathematics. The scope of his theory was perhaps not fully appreciated until the twentieth century. As one implicit detail, Grassmann’s work included the algebra of the tensor calculus that became widely known only after its application (1915-16) in general relativity. The central difficulty that had blocked Hamilton in his attempt to create an algebra of vectors in space of three dimen- sions was the commutative law of * multiplication. His own graphic account of how he saw his wa) round the obstruction in a flash of certainty after much fruitless work being readily acces- sible, we need not repeat it here. But it is well worth thoughtful consideration by all students, especially by those who imagine that mathematical inventions fall into people’s laps from heaven. Before Hamilton succeeded, able men had failed to find the clue to a consistent algebra of rotations and vectors in space. For one, A. F. Mobius (1790-1868, German), who in 1823 had been a pupil 7 of Gauss, took a considerable step toward the desired algebra of four fundamental units in his barycentnc calculus of 1827, a work which Gauss complimented as being composed in the true mathematical spirit. But Mobius was TOWARD MATHEMATICAL STRUCTURE 201 balked by the commutative law of multiplication, which he lacked the daring to reject. However, his new algorithm was of importance in the development of analytic projective geometry, particularly in the use of homogeneous equations, and he was an independent discoverer of the geometric principle of duality.’' So his effort was anything but wasted. The four fundamental units 1, f, /, k of Hamilton’s quarter- nions a + hi -f- cj + dk , (a, b, c, d real numbers), do for rotations and stretches in space what 1, i do for the like in a plane. But whereas multiplication of complex numbers is commutative, that of quaternions is not. Familiar as mathematicians are today with swarms of algebras in which the postulates of common algebra arc severally violated, they can still appreciate the magnitude of Hamilton’s success when, in a flash, he transcended the tradition of centuries. His insight is comparable to that of the founders of non-Euclidcan geometry, or to that of the arithmeticians who restored the fundamental law of arithmetic to the seemingly lawless algebraic integers. It is radical departures from traditional orthodoxy such as these that carry' mathematics forward what seems like a century or more at one stride. The painstaking, detailed cultivation of a newly discovered territory is necessary if it is to be fruitful; but such work can be well done by' mere competence, while radically' new discovery (or invention) is possible only' to men who may imagine they arc conservatives, but who at heart arc rebels. Their boldness may cost them their scientific reputations or the comforts of a decent livelihood; for the way' of the transgressor — who may be only a harmless innovator with the courage to step out in front of the rabble of respectable mediocrity — is some- times as hard in science as it is elsewhere. Grassmann paid for his rashness with eighteen years of obscurity and final scientific extinction for the remainder of his life. Gauss, long in possession of non-Euclidcan geometry, preferred his peace of mind to what he called “the clamor of the Boeotians,” and kept his treasure to himself. Hamilton, having won an imperishable success in optics and dynamics, courted the indifference of his contem- poraries when he devoted all of his superb talents to quaternions and, during his lifetime, acquired exactly one competent disciple in algebra. P. G. Tnit (1831-1901, Scotch) gave up his all in mathematics to follow quaternions. Ten years (1853) after his initial discovery, Hamilton pub- lished his Lectures or. quaUrr.ior.s (64 736 Ixxii pages), in 202 THE DEVELOPMENT OF MATHEMATICS which he showed the utility of quaternions in geometry and spherical trigonometry. But the geometry was Euclidean and of three dimensions. The massive Elements of quaternions (lvii -J- 76 2 closely printed pages) followed in 1866, the year after Hamil- ton’s death. If anything could have convinced geometers and physicists that quaternions were the master key to geometry, mechanics, and mathematical physics that Hamilton antici- pated, his Elements should have done so. Literally hundreds of applications to these subjects were made by Hamilton in this elaborate work, which he considered his masterpiece. Many reasons have been suggested for the failure of qua- ternions to fulfill Hamilton’s expectations. A sufficient explana- tion, which includes many of the others, is that the calculus of quaternions was simply too hard for the busy scientists whom Hamilton would have helped. It took too long to master the tricks. But the possibility of an algebra specifically adapted to Newtonian mechanics and some parts of mathematical physics had been more than merely suggested, and it was reasonably certain that such an algebra would be forthcoming when the need for it became acute. Whatever form the desired algebra might assume, it was also a fair guess that it would follow the example of quaternions and reject the commutative law of multiplication. On the long view, then, the permanent residue of Hamilton’s tremendous labor was the demonstrated existence of a self- consistent algebra in which the commutative law of multi- plication does not hold. This in turn, like the invention of non-Euclidean geometry, encouraged mathematicians to break the iron law of custom elsewhere and to create new mathematics in defiance of venerated traditions. A striking instance, which was to prove of cardinal importance in the development of algebra and the number system, was the construction of algebras in which ab = 0 without either a or b being zero, or in which a n ^ 0 (n = 0, 1, . . .i , w), but a m+1 = 0. A simple instance of the former occurs in hyoolean algebra (belonging to the algebra of logic), in which the ''fact stated is the symbolic expression of Aristotle’s law of contradiction. Linear associative algebra furnishes any desired number of algebras containing 'divisors of zero’ — such as a, b described above — also any number of algebras of the second species. The origin of all these modifica- tions of common algebra is in the work of Hamilton and Grass- mann in the 1840’s. TOWARD MATHEMATICAL STRUCTURE 203 Grassmann’s outlook was much broader than Hamilton’s. To appreciate how much broader it was, we must remember that in 1844, when Grassmann published his first Ausdehnungslehrc, ‘space,’ for all but A. Cayley (1S21-1895, English), was still imprisoned in Euclid’s three dimensions. Cayley’s sketch of a geometry of n dimensions is dated 1843; it could not possibly have influenced Grassmann’s theory of ‘extended magnitude,’ which also can be phrased in the language of ?;-dimensional space. A ‘real’ space or ‘manifold’ of n dimensions is the set, or class, of all ordered n-plcs (.Vi, A'*, . . . , x„) of n real numbers a'i, a*j, . . . , A'„, each of which ranges over a prescribed class of real numbers. It is sufficient for purposes of illustration to let each of x x , a';, . . . , ,v„ range independently over all real num- bers. The class of all (.Vt, x s , . . . , x„) is also called an Jt-dimen- sional real number manifold. In effect, Grassmann associated with (x x , a-j, . . . , x„) the hypercomplex number AVi 4" A';C: 4* * * • 4* x„e n , where e x , c», . . . , e„ arc the fundamental units of the algebra, which he proceeded to construct, of such hypcrcomplcx numbers. Two such numbers, xh*i + ' ‘ + x„e n and y\C\ + • • • + arc defined to be equal if, and only if, a*i = yi, . . . , x n = y„. Addition was defined by (.ViC) 4- • • * + AVn) + (yid + • • • y„fn) - {xj + yj)o + • • • + (•’••„ 4~ yn)fn, of which an instance is common vector addition if n — 2 or if ?: = 3. The various special kinds of multiplication that can be defined at will give the general algebra its chief interest. To demand a definition of multiplication without stating what properties the product is to have is meaningless. If, for example, the associative law a(bc) = ( ab)c is to be preserved, this is equivalent to imposing certain conditions on the funda- mental units Ci, ... , if either of the distributive laws, a{b 4- c) — ab 4" nc, ( b 4* c)a — ba 4 - ca, is to hold, this must be expressed in terms of relations between rj and similarly for the commutative law of multiplication, ab —ba. Thinking partly in terms of geometrical imagery, Grassmann defined several types of multiplication. In particular, multiplying out (ujCs 4 - • - • 4 - a n fn)(bjfi 4- ' ‘ • 4- £ r .<v>)* mid assuming that the ‘coordinates’ a a b u . . . , b n commute with the units e Xi . . . , e n . so that — ajjtc x r t , etc., Grass- mann called e x e u e x e z , c z c h . . . , <r n _ 5 r P ., <v.r n _ j, in the distributed 204 THE DEVELOPMENT OF MATHEMATICS product + <Zi& 2 ^i ^2 + azbie<ie\ + • • • units of the sec- ond order; and he first imposed conditions on these new units. For example, the product of a x e r + • • • and Vi + ‘ • was called an inner product if e r e, = 1 or 0 according as r = s or r ^ s; and an outer product if e T e s = —e s e r , forr, s = 1, . . . n% From these two kinds of products, Grassmann constructed others for more than two factors. For example, if e r je s denotes the inner product of e r , e s , and [ e r e s ] the outer product, there are the possibilities [e r \e s ]e t , e r \[e s e t ], among others, for the definition of products of three factors. A type of particular importance is that in which each of the n 2 products e r e s is a linear homogeneous function of the fundamental units e h ... , e n , and multiplica- tion is postulated to be associative. The linear associative alge- bras_of B. Peirce (1809-1880, U.S.A.), developed in the 1860’s but first printed in 1881, are of this type. 9 A third type of product, called ‘open’ or ‘indeterminate,’ was to prove of central importance in the creation (1881-84) of a practical vector analysis by J. W. Gibbs (1839-1903, U.S.A.). The modern name for such a product is a matrix. 10 Gibbs, one of the most powerful mathematical physicists of the nineteenth century, was perhaps better qualified than Grassmann or Hamil- ton to sense the kind of algebra that would appeal to students of the physical sciences. His most original mathematical contribu- tions in this direction were in dyadics and the linear vector function. These hints must suffice to suggest that as early as 1844, Grassmann was in possession of an extensive theory capable of almost endless developments by specialization in various direc- tions. As elaborated by its creator, this theory of ‘extended magnitudes’ might be interpreted as a greatly generalized vector analysis for space of n dimensions. It incidentally accomplished for any finite number of dimensions what Hamilton’s quaternions were designed to do for Euclidean space of three dimensions. We have already noted that Grassmann’s algebra includes quaternions as a very special case. As a general kind of algebra it also includes the theories of determinants, matrices, and tensor algebra. In short, Grassmann’s theory of 1844-62 was anywhere from ten to fifty years ahead of its epoch. Our present interest in Grassmann’s work is the wide general- ization it afforded of complex numbers X\ + ix-i as number - couples (xi, X 2 ) to hypercomplex numbers {x\, . . . , #„). We must now relate this extension of the number concept to another, TOWARD MATHEMATICAL STRUCTURE 205 made explicitly in 1858 by Cayley but already implicit in the work of Grassmann, namely, matrices. The elements of the theory of matrices are now included in the usual college course in algebra; and since their appearance (1925) in the quantum theory, matrices have become familiar to mathematical physicists. The invention of matrices illustrates once more the power and suggestiveness of a well-devised notation; it also exemplifies the fact, which some mathematicians are reluctant to admit, that a trivial notational device may be the germ of a vast theory having innumerable applications. Cayley himself told Tait 11 in 189-1 what led him to matrices. “I certainly did not get the notion of a matrix in any way through quaternions: it was either directly from that of a determinant; or as a convenient mode of expres- sion of the equations x' = ax + by y’ = ex + dy.” Symbolizing this linear transform variables by the square array ^ meats,’ Cayley was led to his algebra of matrices of n- elements by the properties of linear homogeneous transformations of n independent variables. Behind this invention there is a relevant bit of history. Cayley had shown (1858) that quaternions can be represented as matrices a with a, l, c, d certain complex numbers. To Tait, the pugnacious champion of quaternions ever since he had elected himself Hamilton’s disciple in 185-1, 11 this discover)' of Cayley's was conclusive evidence that Cayley had been inspired to matrices by his master’s quaternions. Because the multiplica- tion of matrices is in general not commutative, and since the like is true of quaternions, therefore, etc. This illustrates the unreliability of circumstantial evidence in mathematics as else- where. But for Cayley's testimony, critics might even now be asserting that Hamilton anticipated Cayley in the invention of matrices, or at least that Cayley got the notion of a matrix from quaternions. Applications, or developments, of these extensions of number followed two main directions. The first, in the geometrical tradition of Hamilton and Grassmann, led to the extremely ation on two independent ^ of its coefficients or ‘cjc- 206 THE DEVELOPMENT OF MATHEMATICS useful vector algebras of classical mechanics and mathematical physics, and later to the tensor algebra and calculus of relativity with its modifications and generalizations in modern differentia) geometry, also to the matrix mechanics of the quantum theory. The second, in the arithmetical spirit of Gauss, guided in part by the abstract algebraic outlook of Galois, led to a partial but extensive arithmetization of algebra. The course of both was highly intricate and blocked by innumerable details, many of which still promise to be of some enduring significance. But to see the principal trends at all, special and strictly limited devel- opments must be ignored, for the present at least; and we shall attend only to the shortest paths from the past to the gains just indicated. From vectors to tensors The line of descent of vector algebra in general is fairly clear. The composition of velocities or of forces in the corresponding parallelogram laws suggested the addition of ‘directed magni- tudes.’ Wessel’s or Argand’s diagram for depicting complex numbers was equally suggestive visually, geometrically, and kinematically. Hamilton and De Morgan’s ‘double algebra’ of number-couples, replacing that of complex numbers, naturally suggested a generalizatidn>.to number-triples, -quadruples, and so on. As we have seen, the central difficulty was the purely algebraic obstacle of commutative multiplication. Thus, in at least the early stages, geometrical and mechanical intuition shared about equally with formal algebra in the creation of a workable mathematics of vectors. The famous Treatise on natural philosophy (1879) of Thomson and Tait offered a magnificent opportunity to display the power of quaternions as an implement of exposition and research in mechanics. Tait exhorted Thomson to repent of his Cartesian sins and embrace the true faith of quaternions. But W. Thomson (Lord Kelvin, 1824-1907, Scotch), declaring that Hamilton’s good mathematics had ended with the masterpieces on optics and dynamics, hardened his heart and persisted in his iniquitous coordinates. The great opportunity was missed. Tait had a somewhat better success with J. C. Maxwell (1831-1879, Scotch). In his epoch-making Treatise on electricity and magnetism (1873, Art. 11), Maxwell made a slightly damning concession: “I am convinced . . . that the introduction of the ideas, as distinguished from the operations and methods of TOWARD MATHEMATICAL STRUCTURE 207 Quaternions, will be of great use . . . especially in electro- dynamics ...” And, with one exception, Maxwell studiously avoided quaternions. The exception (Art, 618) is a summary in quaternion notation of the electromagnetic equations. No use is made of this summary. But Maxwell did use “the ideas,” not of quaternions, but of his own conception of vector analysis. His convergence is the negative of the divergence in use today, and he introduced (Art. 25) what is now called the curl of a vector. These innovations have lasted. The most profitable departure from quaternionic orthodoxy was that of J. W. Gibbs in his vector analysis of the ISSO’s. This will be noted presently. The next was by O. Heaviside (1850- 1925, English), in his profoundly individualistic Electromeig7ietic theory of 1893. In a chapter of 173 pages, Heaviside elaborated Ins own vector notation. His methods resembled those of Gibbs; but of Gibbs’ notation, Heaviside confessed, “I do not like it.” Germany provided the next (1S97) considerable variation on the now familiar theme, in A. Foppel’s Gcovictrie der JVirb elj elder — geometry of vortex fields. By 1900 the contest between rival claimants to physical favor had narrowed down in English- speaking countries to Gibbs versus Heaviside. Quaternions appeared to have been knocked out. Tait, their most formidable champion, died in 1901; the vector analysis of Gibbs or some modification of it prevailed in the U.S.A. Much of this tortuous development was enlivened by one of the most spirited mathematical controversies of modern times. Unlike the numerous squabbles over priority, the quaternions- vcrsus-vcctors war was refreshingly scientific. The casus belli was a purely mathematical difference of opinion: were quater- nions a good medicine for applied mathematics, or was some one of several diluted substitutes a better? The uninitiated might think that so abstract a bone of contention would provoke only dry academic discourse, with at worst an occasional growl of dissent. It did nothing of the kind. The language of the dis- putants even bordered on quite un-Victorian indelicacy at times, as when Hamilton’s devoted Tait ,: in 1890 called the vector analysis of Gibbs “a sort of hermaphrodite monster, com- pounded of the notations of Hamilton and Grassmann.” That was Scotch and Irish against American. Gibbs, being a New Englander to the marrow and a confirmed bachelor cherished only by his married sister, was but slightly acquainted with the inexhaustible resources of the American language. Tait got away 208 THE DEFELOPMENT OF MATHEMATICS with his abnormal physiology, but Gibbs got the better of the mathematical argument. Frenchmen, Germans, and Italians, urging their respective substitutes for quaternions, added to the din. By the second decade of the twentieth century there was a babel of conflicting vector algebras, each fluently spoken only by its inventor and his few chosen disciples. If, at any time in the brawling half- century after 1862, the bickering sects had stopped quarreling for half an hour to listen attentively to what Grassmann was doing his philosophical best to tell them, the noisy battle would have ended as abruptly as a thunderclap. Such, at any rate seems to have been the opinion of Gibbs. In retrospect, the fifty- year war between quaternions and its rivals for scientific favor appears as an interminable sequence of duels fought with stuffed clubs in a vacuum over nothing. The disputes ceased to have any but a mathematically trivial significance almost as soon as they began. As Gibbs 10 emphasized in 1886, in his account of the development of multiple algebra, the mathematical root of the matter is in Grassmann’s indeter- minate product, that is, in the theory of matrices. Gibbs also remarked the superior generality of Grassmann’s many possible kinds of product in multiple algebra over the unique product insisted upon by Hamilton: Given only the purely formal law of the distributive character of multipli- cation — that is sufficient for the foundation of a science. Nor will such a science be merely a pastime for an ingenious mind. It will serve a thousand purposes in the formation of particular algebras. Perhaps we shall find that in the most important cases the particular algebra is little more than an application or interpretation of the general. The whole of Gibbs’ judicial and profound evaluation (1886) of multiple algebra in relation to its applications might be studied with profit at any time by those interested in the con- tinued improvement of applied algebra. Vector analysis and even the infinitely more inclusive Ausdehnungsleliren of Grassmann are after all only provinces, although highly cultivated ones, of algebra, which itself is but a territory of modern mathematics. Those interested in the advancement of mathematics, rather than in the perpetuation of individuals as dictators of provinces, will not be dismayed when particular theories to which they may be personally attached are supplanted by others. Obsolescence is a necessary adjunct of progress; and any effort such as Tait s to keep quaternions unsullied and perpetually fresh is likely to be TOWARD MATHEMATICAL STRUCTURE 209 as futile as an attempt to stop the earth in its orbit. The vector analysis of Gibbs gradually displaced quaternions as a practical applied algebra in spite of the utmost efforts of the quaternion- ists; and after 1916 it seemed that the several special brands of vector analysis were about to be supplanted in their turn by the tensor algebra and analysis that became popular in 1915-16 with the advent of general relativity. As in the struggle of vector analysis against quaternions, the advance to tensors generated its own opposition. Vector analysis, like some human beings, needed above all else to be delivered from the good intentions of its partisan friends. Progress here, as elsewhere in the past of mathematics, appeared to be possible only when all the friends and former pupils of some great and justly famous master should have died. Then only might it be possible to see the mathematics rather than the man. Such retardations due to misdirected enthusiasm are frequent enough in mathematics. The master founds a ‘school’; the pupils, remembering perhaps among other things an encouraging pat on the head from their first competent teacher, graduate into a world that docs not stop dead no matter who dies, to keep on repeating for the rest of their lives the only lesson they ever really learned. The school itself expires, leaving its useful con- tribution encrusted with an accumulation of artificially stimu- lated growths that must be cut away before the creative idea of the originator can begin to live and function freely. Aware of these possibilities, some mathematicians, including one of the first rank, have refrained from propagandizing their own ideas or those of their teacher, and have made no attempt to gather a following of bigoted disciples. Kronecker took pride in the fact that he had never tried to found a school or to acquire a host of students. He believed, as did Gibbs, that “the world is too large, and the current of modern thought is too broad, to be confined by the ipse dixit even of a Hamilton.” There seems to be but little doubt that applied algebra was held back by the partisans of jealous schools. The road to unity can be traced back from about 1940, when the rudiments of the tensor calculus had become fairly common in undergraduate instruction, to Grassmann’s n-dimensional manifolds of 1844. Three dimensions arc inadequate for modern physics, or even for classical mechanics with its generalized coordinates. G. F. B. Ricmann (1S26-1866, German) in 1854 took the next long step forward after Grassmann when he introduced Gaussian (in- 210 THE DEVELOPMENT OF MATHEMATICS trinsic) coordinates and made 72 -dimensional manifolds basic in his revolutionary work on the foundations of geometry. Another work of Riemann’s, published after his death, contained what is now known as the Riemann-ChristofFel tensor in the rela- tivistic theory of gravitation. Riemann encountered this tensor in a problem on the conduction of heat. E. B. Christoffel (1829— 1900, German) was the nest to make significant progress toward a general tensor calculus, in his work of 1869 on the transforma- tion (equivalence) of quadratic differential forms. Finally, in the 1880’s, the Italian geometer M. M. G. Ricci combined and added to all the work of his predecessors. The result, published in 1888, 13 was the tensor calculus. Thus the mathematical machinery demanded by the theory of general relativity was available a year after the Michelson-Morley experiment, which was partly responsible for the special theory of relativity in 1905; without the tensor calculus the general theory' of 1915-16 would have been impossible. The above assertion about the Michelson-Morley experiment does not imply that Einstein was motivated by' the experiment in his construction of special relativity. In fact he has stated explicitly* that he knew of neither the experiment nor its outcome when he had already convinced himself that the special theory was valid. The new method attracted very little attention. On the invitation of F. Klein (1849-1925, German), Ricci and his former pupil, T. Levi-Civita 1873-1942, Italian), prepared an article on the tensor calculus and its applications to mathematical physics for publication in a journal read by* mathematicians of all nationalities. The article, in French, appeared in 1901. It fell rather flat. However, a few curious geometers outside of Italy became aware of the new calculus, and at least one, M. Gross- mann of Zurich, mastered it and taught it to Einstein. The tensor calculus was the particular kind of generalized vector algebra appropriate for expressing the differential equations oi relativity in covariant form as demanded by a postulate of tne theory. The debt of algebra and geometry to general relativity is 2 s great as that of relativity to algebra and geometry. Although Ricci and Levi-Civita in their expository article of 1901 had offered abundant evidence of the utility of tensor analysis in applied mathematics, the new calculus was seriously taken up by mathematical physicists only after their curiosity had been roused by the experimentally verified mathematical predictions TOWARD MATHEMATICAL STRUCTURE 211 of relativity. The tensor method quickly induced a vast develop- ment of differential geometry. Gibbs had predicted in 1886 that vector analysis would some- day greatly simplify what in his time was modern higher algebra — the theory of algebraic covariants and invariants. He had in mind the possibilities of Grassmann’s theory. His prediction was verified in the 1930’s. Another prediction of Gibbs of the same kind was verified in 1925, when W. Heisenberg found in the algebra of matrices the implement he needed for the non-com- mutative mathematics of his quantum mechanics. Physicists took less kindly to ab ^ ha than they had to tensors; and it was a great relief to many when C. Eckart (U.S.A.) and E. Schrodinger (Austria) in 1926 showed independently and simul- taneously that matrix mechanics could be replaced by wave mechanics, in which the theory of boundary-value problems, already familiar in classical mathematical physics, is the key to the mathematics. It scents probable that Grassmann did not anticipate any such outcome for his extremely general ‘geometrical algebra.’ Two of his successors, Riemann and W. K. Clifford (1845-1879, English), both more physical-minded than Grassmann, ventured to predict the twentieth-century geometrization of some parts of mathematical physics. This was in the middle stage from Grassmann to tensors, and it was as remarkable a prophecy as any that mathematicians have ever made. But it must not be forgotten that mathematicians no less than scientists and others have made many false prophecies. The successes are remembered. Toward structure “Mathematics,” according to Gauss in 1831, “is concerned only with the enumeration and comparison of relations.” B. Peirce (1809-1880, U.S.A.), one of the creators of linear associa- tive algebra, asserted 9 in 1S70 that “Mathematics is the science which draws necessary conclusions.” Peirce also remarked that “all relations arc either qualitative or quantitative,” and that the algebra of either kind of relation may be considered inde- pendently of the other, or that, in certain algebras, the two may be combined. These opinions, from what is now a remote past mathe- matically, might be admitted by some formalists as anticipations of their own conception of mathematics as the theory of struc- 212 THE DEVELOPMENT OF MATHEMATICS ture. In particular, the Pythagorean program is superseded. Euclid’s postulational method remains. Large tracts of mathe- matics have become entirely formal and abstract; the content of a mathematical theory is the structure of the system of postu- lates from which the theory is developed by the rules of mathematical logic, and from which are derived its various interpretations. This excessively abstract view of mathematics evolved from the formalization of elementary algebra in the 1830’s, which has already been described; the work of Abel and Galois in the theory of algebraic equations, of about the same time; the development of linear algebra throughout the nineteenth and early twentieth centuries; the creation of mathematical logic, beginning with Boole in 1847-54 but vigorously pursued only in the twentieth century; and finally, from the free invention of non-Euclidean geometries after 1825, and the renewed interest in postulational methods following Hilbert’s work of 1899 on the foundations of geometry. Of all these influences, two in particular are germane here: the development of linear algebra; and the infiltration of the ideas of Abel and Galois into algebra as a whole. The Galois theory of equations was acknowledged by both Dedckind and Kronecker to be the inspiration for their own general and semi- arithmetical approach to algebra. Two of the basic concepts of the Galois theory, domains of rationality, or fields, and groups, were the point of departure. Both groups and fields will be described presently. For the moment we observe the underlying methodology -which might have been followed and which would be followed today (1945), but which vras not followed histori- cally, in the generation of linear algebras, groups, and other systems in modern algebra. The methodology is that of generalization by suppression of certain postulates defining a given system. The system defined by the curtailed set of postulates is then developed. Linear algebra is obtainable in this way from the algebra of a field. Vector algebras, as we have seen, received their initial impulse from Hamilton’s suppression of the postulate that multiplication is commutative in common algebra. Common algebra is the most familiar example of a field. Groups also may be derived from common algebra oy tne same technique of generalization. But they were not so obtain/— originally: and it is doubtful whether they would ever h~ ,c TOWARD MATHEMATICAL STRUCTURE 213 attracted the attention they did, had not the momentum of history thrust them forward. There are 4,096 (perhaps more) possible generalizations of a field. To develop them all without some definite object in view would be slightly silly. Only those that experience has suggested have been worked out in any detail. The rest will keep till they are needed; the apparatus for developing them is available. Nevertheless, the postulational technique has been one of the most suggestive of twentieth- century mathematics; and we shall have occasion to recur to it frequently as we proceed. Fields being the most familiar of all mathematical systems, we shall define them first. A field 14 ( Korper , corpus, corps, do- main of rationality) F is a system consisting of a set S of elements a, b, c, s, . . . and two operations, ©, O, which ma_v be performed upon any two (identical or distinct) elements a, b of S, in this order, to produce uniquely determined elements a © b and a O b of S, such that the postulates (1) to (5) arc satisfied. Elements of S will be called elements of F. For sim- plicity, a © b, a G b will be written a -j- b, ab. (1) For any a, b of F, a + b and ab are uniquely determined dements of F, and b + a — a + b, ba = ab. (2) For any a, b, c of F, (a + b) + c = a + (b -f c ), ( al>)c — a{bc), a(b -f c) = ab -{- ac. (3) There exist in F two distinct elements z, v such that if a is any element, a + s = a, au — a. (4) For any element a of F, there is in F an dement .v such that a 4- x = z. (5) For any clement a, other than z, of F, there is in F an dement y such that ay = v. It should be noticed that equality, =, has been assumed as a known relation. For completeness: equality is an equivalence relation (as defined earlier in connection with congruence). That is, if a, b arc any dements of F, a — b or a ^ b, ^ meaning 'not equal to’; a — a; if a = b, then b — a- if a = b and b — c, then a — c. This familiar and somewhat elaborate abstraction of common algebra and rational arithmetic will serve to illustrate the mean- ing of structure and the history of its development. We note first that these precise postulates date only from 1903; and that in the postulates as given in 1903 (and 1923), the precise meaning of cqtialitv is not stated, being taken for granted. In texts of 1930 or later, it became customary to define equality as an 214 THE DEVELOPMENT OF MATHEMATICS equivalence relation before using equality in the postulates of a field. This is typical of the continually increasing precision in elementary mathematics since the first explicit definition of a number field in 1879 by Dedekind. As a final instance of the same tendency, it was only in the 1920’s that it became cus- tomary to state explicitly that a = b or a ^ b. There is therefore little reason to suppose that even these precisely stated postu- lates have explicated all the assumptions underlying our habitual use of common arithmetic. If, in the postulates (1) to (5), the elements, a, b, c, . . be interpreted as rational numbers, and u, z as 1, 0, with a + b, ah the sum, product of a , b, it is seen that the rational numbers are an instance of a field with respect to addition and multiplication. Subtraction and division follow from (4), (5). Similarly, ordinary complex numbers x + iy furnish another instance; as also do Hamilton’s number-couples (#, y) with the appropriate defini- tions of u, z, addition, and multiplication, which the reader may easily recover. The rational integers 0, ±1, +2, ... do not furnish an instance, on account of (5). If F is any field, and Xi, . . . , x n are independent variables (or indeterminates), the set of all rational functions of Xi, . . . , x n , with coefficients in F, is another field. With ‘structure’ still not defined formally, it is intuitively evident what is meant by the statement that all instances of a field have the same structure, and that this structure is as in (1) to (5). Further, it is clear that if the logical consequences of (1) to (5) are developed, the body of theorems so obtained will be valid for each instance of a field. The last is indeed ‘clear,’ although a proof of it might be difficult and, as a matter of fact, no generally accepted proof had been devised up to 1945. A thoroughly satisfactory proof must demonstrate that the rules of mathematical logic applied to (1) to (5) will never produce a contradiction, such as “a = b and a ^ b.” It seems as if this must be the case; but seeming in mathematics is not the same as being. ‘Existence,’ for one school, is indeed identified with proof. The earliest recognitions of fields, but without explicit defini- tion, appear to be in the researches of Abel 15 (1828) and Galois 16 (1830-1) on the solution of equations by radicals. The first formal lectures on the Galois theory were those of Dedekind to two students in the early 1850’s. Kronecker also at that time began his studies on abelian equations. It appears that the con- cept of a field passed into mathematics through the arithmetical TOWARD MATHEMATICAL STRUCTURE 215 works of Dedekind and Kronecker. Both, especially Dcdekind, ,<! early recognized the fundamental importance of groups for algebra and arithmetic. With Dedekind's famous Eleventh supplement to the third edition (1S79) of P. G. L. Dirichlet’s (1S05-1S59, German) J'orlesungen uber Zahier.tkeorie , the con- cept of a number field was firmly established in mathematics. We note, however, that Dedekind in this work was interested only in algebraic numbers — roots of algebraic equations with rational number coefficients. The fields he defined were therefore those of real and complex numbers. Kronecker followed in 18S1 with his domains of rationality, that is, fields. Although Kroncckcr’s definition was more general than Dedekind's, it did not attain the complete generality of the postulate system quoted above. The passage to final abstractness took about a quarter of a century. This need not be traced in detail here; the references given are sufficient to orient anyone who wishes to elaborate the history. The turning point was Hilbert’s work on the foundations of geometry in 1S99. Although this did not concern algebra or arithmetic directly, it set a new and high standard of definiteness and completeness in the statement of all mathematical defini- tions or, what is equivalent, in the construction of postulate systems. Compared to what came after 1 900 in this basic kind of work, that before 1900 now seems incredibly slack. With abun- dant resources at hand to continue the Euclidean program of stating explicitly what a mathematical argument is to be about, a majority of nineteenth-century mathematicians left their readers to guess exactly what was postulated. Neglect to state all the intended assumptions incurred its own penalties in faulty proofs and false propositions. The change for the better after 1900 was most marked, but there is still room for improvement, especially in mathematics of the intuitive kind — such as the repeated appeal thus far to intuition for the meaning of structure. Passing to groups, we shall state in full a set of postulates for a group, as ‘group’ in the technical sense defined by these postulates will occur repeatedly in the sequel. We shall then be in a position to define structure. A group G is a set 5 of elements a. b, c, . . . , x, y, . . . and an operation 0 , which may be performed upon any two (identical or distinct) elements n, b of S, in this order, to produce a uniquely determined element cOb of S, such that the postulates (1) to (5) are satisfied. (1) c.0b is in S for every a, b in S. 216 THE DEVELOPMENT OF MATHEMATICS ( 2 ) aOibOc ) = ( aOb)Oc for every a , b, c in S. (3) For every a, b in S there exist x,y'mS such that aOx = b, yOa = b. These postulates may appear strange to those acquainted with others for a group; but they are simpler than some, and all are equivalent. Historical notes on groups will be given later; our present interest is in mathematics. We proceed to structure, 19 which seems to have been first recognized, but not defined, in groups. Consider two groups, with the respective elements a h b h c i, . . . and az, bz, c 2 , . . . and the respective operations Oi, 0 2 . These groups are said to be simply isomorphic, or to have the same structure, if it is possible to set up a one-one corre- spondence between the elements such that, if XiOiyi = Z\, then x 2 0 2 yz = Zz, and conversely, where x h y if Zi are the respective correspondents of x 2 , yz, z 2 . For further details we must refer to the texts. This definition is probably the simplest example of what is meant by ‘same structure.’ Note that ‘structure’ is not defined, but that ‘same structure’ is. For the purposes of algebra this is sufficient. If ‘same structure’ seems at first glance to define absolute identity, an example to the contrary is supplied by all the normal men in a community, all of whom have the same shape — two arms, one head, etc. — but no two of whom are identical except perhaps topologically. A general theory of structure was developed by A. N. White- head ( 1861 -, English) and B. Russell 21 (1872 — , English) in 1910 . It will suffice here to recall a cardinal definition: A relation P between the members of a set x p has the same structure as a relation Q between the members of a set y q if there is a one-one correspondence between the elements of x v and y q such that, whenever two elements of x v are in the relation P to each other, their correlates (by the correspondence) in y q are in the relation Q to each other, and vice versa. If in any division of mathematics there are relations P, Q, . . . having the same structure, it suffices to elaborate the implications of one, say P, when those of Q, . . . follow on translating from P, x p , . . . to Q, y q , . . . by means of the relevant correspondence in each case. Each of the postulates of a mathematical system can be restated as a relation between the data (‘elements’ and ‘operations’) of the system. If it is possible to establish a one-one correspondence between TOWARD MATHEMATICAL STRUCTURE 217 the postulates of two systems such that correlated postulates have the same structure, then the systems are said to have the same structure. Instead of saying that two systems have the same structure, it is customary in the U.S.A., following E. H. Moore (1862-1932, U.S.A.) who used the concept in his lectures and writings from about 1893 on, to say that the systems are abstractly identical. Abstract identity is itself an equivalence relation. If several systems are abstractly identical, obviously it is sufficient to develop the mathematics of one in order to have that of all. The systems so developed will differ in the interpretations assigned to the abstract elements and operations; each assignment provides an ‘instance’ of the theory. For exam- ple, the algebras of real and of complex numbers, or of Hamil- tonian number-couples, are instances of the theory of an abstract field. Tracing the evolution of algebra since the 1830’s, we note a constant but largely subconscious striving toward abstractness. Concomitantly, abstract identity was sought, sometimes delib- erately, as in the theories of groups and fields. Most classification is an effort in the same direction preparatory to comparison of different theories and the detection of abstract identities. Klein’s unification of diverse geometries by the theory of groups in 1872, which will be described in connection with invariance, was a conspicuous example of the advantages accruing from a recog- nized abstract identity. But it seldom happens that anything so simple as a group unifies apparently unrelated divisions of mathematics with respect to anything deeper than superficialities. Enough has been said about structure to indicate what Gauss may have had in mind when he observed that “mathematics is concerned with the enumeration and comparison of relations.” He made this statement in connection with complex numbers. On another occasion he expressed a doubt that any ‘numbers’ other than the real and complex — such as quaternions, for example — would ever be of any use in the higher arithmetic. In our further pursuit of algebra and arithmetic we shall be guided by this hint of Gauss’, and endeavor to sec what he might have had in mind. This, of course, is not the only road by which number might be followed from the lS30’s to the twentieth century. But following it, we shall have a definite object in view by which to orient some of the major trends of algebra and arithmetic on the way. CHAPTER 10 Arithmetic Generalized Continuing with the modern developments of number and their influence on the emergence of structure, we shall observe next the expansion of modern arithmetic — the Greek arithmetica — from its origin in 1831 in the work of Gauss on the law of biquadratic reciprocity to its end in mathematical logic. Our immediate interest in this chapter is the greatly generalized concept of whole number, or integer, which distinguished the higher arithmetic of the late nineteenth century from all that had preceded it. In a subsequent chapter we shall follow some of the main lines of descent of the classical arithmetic from Fermat, Euler, Lagrange, and Gauss to the present. Historically, many of these older developments preceded the work to be described here. But their interest, great though it may be intrinsically, is as yet comparatively negligible for mathematics as a whole. There are six major episodes to be observed, four of which will be described in this chapter and the following. The four are the definition by Gauss, E. E. Kummer (1810-1893, German), and Dedekind of algebraic integers; the restoration of the funda- mental theorem of arithmetic in algebraic number fields by Dedekind’s introduction of ideals; the definitive work of Galois on the solution of algebraic equations by radicals, and the theory of finite groups and the modern theory of fields that followed; the partial application of arithmetical concepts to certain linear algebras by R. Lipschitz (1831-1903, German), A. Hurwitz (1859-1919, Swiss), L. E. Dickson (1874-, U.S.A.), Emmy Noether (1882-1935, German), and others. All of these develop- ments are closely interrelated. The last marks the farthest extension of classical arithmetic up to 1945, and is either the climax or the beginning of a structural arithmetization of algebra 218 ARITHMETIC GENERALIZED 219 foreseen as early as 1860 by Kroncckcr, but only partly achieved by him in the 1880’s. As if in preparation for the climax, the algebra of hvpercompicx numbers rapidly outgrew its classifica- tory adolescence of the lS70’s, represented by the work of B. Peirce and his successors, and became progressively' more con- cerned with general methods, reaching a certain maturity' early' in the twentieth century'. The fifth major episode, which logically' would seem to be a necessary' prelude to the others, strangely' enough came last. Not until the closing y'cars of the nineteenth century' was any'onc greatly perturbed about the natural numbers 1, 2, 3, ... . All mathematics, from the classical arithmetic of Fermat, Euler, Lagrange, A. M. Legendre (1752-1833, French), Gauss, and their numerous imitators, to geometry' and analysis, had ac- cepted these speciously simple numbers as ‘given.’ Without them, none of the major advances of modern arithmetic would ever have happened. Yet no arithmetician asked, “By' whom are the natural numbers ‘given’?’’ Kronecker ascribed them to God, but this was hardly a mathematical solution. The question arose, not in arithmetic, but in analysis. It was answered by the modern definition of cardinal and ordinal numbers. This finally united arithmetic and analysis at their common source. The sixth and last major episode in the evolution of the num- ber concept was the application of arithmetic to the differential and integral calculus. It is a point of great interest, as will be seen in a later chapter, that one of the strongest initial impulses for the final application of arithmetic to analysis came from mathematical physics. Fourier’s theory of heat conduction (1S22) disclosed so many' unforeseen subtleties in the concepts of limit and continuity that a thorough overhauling of the basic ideas of the calculus was indicated. Many toiled at this for the rest of the nineteenth century. It was gradually perceived that the cardinals and ordinals 1, 2, 3, . . . demanded clarification. By’ 1902 the last subtle obscurity then uncovered had been removed, only' to make room for a yet more subtle. The arith- metic of 1, 2, 3, ... , and with it mathematical analysis, resigned its soul to the searching mercies of mathematical logic. About twenty-five centuries of struggle to understand num- ber thus ended where it had begun with Pythagoras. The modern program is his, but with a difference. Pythagoras trusted 1, 2, •’>•*• to ‘explain’ the universe, including mathematics; and the spirit animating his ‘explanation’ was strict deductive rca- 220 THE DEVELOPMENT OF MATHEMATICS soiling. The natural numbers are still trusted by mathematicians and scientists in their technical mathematics and its applications. But mathematical reasoning itself, vastly broadened and deep- ened in the twentieth century beyond the utmost ever imagined by any Greek, supplanted the natural numbers in mathematical interest. When, if ever, mathematical logic shall have surmounted its obscurities, the natural numbers may be clearly seen for what they ‘are.’ But there will always remain the possibility that any unsealed range may conceal a higher just beyond; and if the past is any guide to the future, arithmeticians will come upon many things to keep them busy and incompletely satisfied for the nest five thousand years. After that, perhaps, it will not matter to anyone that 1, 2, 3, . . . ‘are.’ Generalized divisibility The class of positive rational integers 1, 2 ,3, . . . was first extended, as a class of integers , by the adjunction of zero and the negative rational integers —1, —2, —3, .... We recall that Euclid in the fourth century b.c. proved one of the cardinal theorems concerning positive rational primes: If a prime p divides the product of two positive rational integers, p neces- sarily divides one of them. A rational prime admits as divisors only itself and the units 1, —1. The extension of Euclid’s theorem to all the rational integers is immediate and need not be recalled. But to emphasize the non-trivial character of the generalizations of the rational integers by Gauss, Kummer, Dedekind, and others, the preceding definitions must be reformulated so as to apply to the generalized ‘integers’ in question. It may be re- marked that this simple recasting of the definitions of rational arithmetic was one of the three most difficult steps toward the desired generalization. The other two were a redefinition of arithmetical divisibility, as distinguished from division in alge- bra, and the closely related problem of selecting from a given class of numbers those which are to be defined as integers. First, as to units. With ‘integer’ as yet unspecified, a unit In a given set of integers is an integer that divides each integer in the set. An integers ‘divides’ an integer ft if there is an integer y such that ft — erf. Second, as to ‘irreducibles.’ An integer oc is said to be irreduci- ble if l a = ftyft with ft, y integers, implies that one of ft, 7 is a unit and the other is oc. ARITHMETIC GENERALIZED 221 Third, as to primes. An integer a is called a prime if it is irreducible, and if further the assertion ‘ a divides fiy' implies at least one of the assertions ‘a divides /3,’ ‘a divides 7 .’ These definitions accord with those for the rational integers. But whereas rational primes and rational irreducibles coincide, the like is not true for all of the generalized integers to be described. The manufacture of definitions is likely to be a profitless pursuit unless there is a definite end in view. The goal here is the fundamental theorem of arithmetic: the ‘integers’ defined are to be resolvable into powers of distinct ‘primes’ in one way only, apart from ‘unit’ factors and permutations of the factors. This requirement is too drastic for the ‘arithmetic’ of most linear algebras; it is that at which the founders of the theory of alge- braic numbers aimed. It was to prove unattainable. The means by which the original program was replaced by another, which accomplished the essentials of what had been sought originally, is one of the finest examples of generalization in the history of mathematics. The generalization concerned the fundamental concepts of common arithmetic, particularly ‘integer’ and ‘divisibility . 5 To be of more than trivial significance, any generalization in mathematics must yield on appropriate specialization all the instances from which the generalization proceeded, and must give in addition more than is contained in all of those special instances. The profoundest generalizations appear to be those in which the interpretations of all the symbols in the structure (postulates) of a given system arc changed. The passage from rational integers to algebraic integers was of this kind. For example, in the theorem of rational arithmetic, “if a divides b, then b docs not divide a unless < 7 , b arc units,” a , b, (b -A- (T), and the division-relation are all assigned interpretations in the generalization differing from those of rational arithmetic. But these interpretations arc such that the statement “if a , etc. 5 ’ remains true for the new interpretations. Tiie extension of rational arithmetic to an arithmetic of alge- braic numbers and, considerably later, to a partial arithmetiza- tion of linear algebra, originated in two distinct sources: the proof by Gauss in 1828-52, or earlier, of the law of biquadratic reciprocity: Rummer’s attempt in the 1810’s to prove Fermat’s last theorem. We begin with Gauss. If there is a rational integer x such that, when f, q are given positive integers, x* — <7 is divisible (without remainder) 222 THE DEVELOPMENT OF MATHEMATICS by p, q is called an rAc residue of p. Restated as a Gaussian congruence, q is an rAc residue of the given p if and only if 9r = q mod p is solvable for For simplicity, we describe onlv the case in which p. q are positive odd primes. Gauss was particu- larly concerned with r. = 4, r. = 3. For r. — 2. Legendre’s lav of quadratic reciprocity', which Gauss called “the gem of arith- metic,” is (p\q')(q\p) = (-i) i{ ™ 5 -» ■where (p\q) denotes 1 or — 1 according as x- ss p mod q is. or is not, solvable for and similarly for (o;P) and x- = g mod p. Gauss long sought a reciprocity law for r, = 4 as simple as that for r, = 2. He found it only when he passed beyond rational integers to complex integers. A Gaussian complex integer is a number of the form a -f- hi. where a , b are rational integers. De- fining units, primes, and divisibility for his complex integers in the straightforward way suggested by analogy with rational integers. Gauss proved that the fundamental theorem of arith- metic holds for integers a hi. By means of these integers he was enabled to state the law of biquadratic (n = 4) reciprocity concisely. For n — 3 he found an equally simple theory, based on "'integers’ a -f- bp , where p is a root of y- -f y t 1 = 0 and a, b are rational integers: but he did not publish his results. The history of reciprocity laws for r. > 4 would fill a large book. This highly developed subject has been cultivated by scores of arithmeticians, and it has had a considerable influence on the evolution of modem algebra. But as this specialty, rich though it may be intrinsically, is rather to one side of the prin- cipal advance, we must leave it here with a remark. What is essentially the law of quadratic reciprocity was known to L. Euler (1707—1783, Swiss) in 1744—6 but was not proved by him. 1 He discussed the law more fully in 1783. Legen- dre in 1785 attempted a proof, but slipped in assuming as obvious a theorem which is as difficult to prove as the law itself. Gauss first published a proof in 1801, and gave six in all. For r. > 2, the reciprocity' laws depend upon the algebraic number fields enter- ing through binomial equations of degree r,. This brings us to the next stage in the development of algebraic numbers. A particular algebraic number field of degree n is the set ot all rational func- tions of a root of a given irreducible algebraic equation ox degree r. with rational integer coefficients. 2 In his attempt to prove the impossibility which x, y. z. p are rational integers, xyz A of -f m 0, and p is a prime ARITHMETIC GENERALIZED 223 >2, Kummcr in 1849 resolved x p 4* y p into its p linear factors, (x -f y) (x -{- ay) ' ' ' (.v + a J>-1 y), v.'here a is an imaginary pth root of 1. This led him to extend the theory of Gaussian complex integers to the algebraic number field defined by a p ~ i 4* cc p ~ : 4* * * * 4* a -f 1 = 0. With ap- propriate definitions of integers, primes, etc., in this field, Kummcr persuaded himself for a time that he had proved Fermat’s last theorem. But, as P. G. L. Dirichlct (1805-1859, German) pointed out to him, he had assumed that the funda- mental theorem of arithmetic holds for these integers constructed from a. For certain primes p the fundamental theorem is valid in the corresponding a-field; for others it is not. The complete proof (or a disproof) of Fermat’s theorem was still open. Undaunted by this totally unforeseen failure, Kummcr invented a new kind of number, which he called ‘ideal’ — not to be confused with Dedekind’s ideals. There would be no point in describing these here, 3 as they are too far off the main road. They apply to the particular number fields considered by Kummcr in connection with Fermat’s last theorem. Making a completely fresh start in the early 1870’s, J. W. R. Dedekind (1 S3 1-1916, German) created a theory of algebraic integers applicable to the general case of an algebraic number field defined by a root of an irreducible equation r.oX n -p 4~ " * ‘ +d a = 0 of any degree r. with rational integer coefficients a 0 , . . . , a n . A root of this equation is called an algebraic number of degree r.\ if Co = 1, this number is an algebraic integer; if in addition a n = 1 (or —1), the algebraic integer is a unit. Note that any rational integer r is an algebraic integer of degree 1, since r is the root of x — r = 0. All this detail has been recalled to indicate that the generali- zation from rational integers and rational units to algebraic integers of any degree demanded unusual insight. At first glance it seems impossible that a number such as ( — 13 -f- \ — 1 1 5)/2 should have any of the divisibility properties of a common whole number. This specimen, being a root of the irreducible equation .v : -f 1 3x -r"l — 0, is in fact an algebraic integer of the second degree. Algebraic number holds in which there is unique decomposi- tion of algebraic integers into primes arc the exceptions. To 224 THE DEVELOPMENT OF MATHEMATICS restore the fundamental theorem of arithmetic to the integers of any algebraic number field, Dedekind reexamined divisibility for the rational integers. This was the critical step, leading to the invention of what Dedekind called ideals. An (integral) ideal of an algebraic number field A is a subset, say a, of all the integers of F such that, if a, ft are in a, and £ is any integer in F, then a — (3 and a£ are in a. The ideal a is said to divide the ideal b if every integer in b is also in a, that is, if a, considered as a class, contains b. The unit ideal is the set of all integers of F; it divides every ideal. An ideal $ is prime if, and only if, p and the unit ideal are the only ideals dividing p. Unique factorization was restored thus — ‘replaced’ would be more strictly accurate. If a is any integer of F, the set of all ar£, where £ runs through all integers of F, is easily seen to be an ideal. This ideal, denoted by (a), is called the principal ideal corresponding to a; and it follows immediately from the defini- tions that, if a, (3 are any integers of F, a divides /3 when, and only when, (a) divides (/3). ‘Divides,’ in l a divides /3, 5 means that there is an integer 7 of F such that (3 = ay; ‘divides,’ in l {a) divides (/3),’ means that the principal ideal (a) contains the principal ideal (/3); and the theorem asserts that each of these division-relations implies the other. In rational arithmetic, for example, ‘3 divides 12’ is equiva- lent to ‘the class of all integer multiples of 3 contains the class of all integer multiples of 12.’ Again, if a, b are given integers, the class of all integer multiples of a contains the class of all integer multiples of b if and only if a divides b. Decomposition of an algebraic integer into a product of algebraic integers is now mapped onto a decomposition of an ideal into a product of ideals. The fundamental theorem of arithmetic is valid in the map. The mapping is as follows. The integers d, (3 of F are replaced by their corresponding principal ideals (a), ((3), . . . . Since the capital theorem of Dedekind’s theory establishes the unique decomposition of any ideal (in F) into a product of powers of prime ideals, each of (a), Q3), . . . has such a unique decomposition . 4 Roughly, the crux of the matter is the replacement of the relation of arithmetical divisibility by the relation of class- inclusion as in either classical or symbolic logic. And, still roughly, this replacement is in part responsible for the appear- ance of ideals as linear sets of particular kinds in modern algebra and in algebraic geometry. ARITHMETIC GENERALIZED 225 The invention of ideals has been given what may seem more than its legitimate share of space because it is an admirable and easily described example of the modern tendency to generaliza- tion. The most characteristic detail, possibly, is that of replacing the familiar concept of arithmetical divisibility by another that includes it. A central relation is replaced by another bearing no superficial resemblance to the first. Nevertheless, after the replacement, a cardinal theorem (unique decomposition into primes) is restored, by mapping or one-one correspondence, to a domain in which, before the replacement, the theorem did not hold generally. And further, the replacement leaves essentially unaltered those cases in which the theorem held before the replacement. From another point of view, the replacement of the set of algebraic integers by the set of correlated principal ideals intro- duces uniformity and brings apparent anomalies under a new and wider law. An earlier instance of the same procedure oc- curred in the introduction of ideal elements (points, lines, planes, ... at infinity) into projective geometry during the first half of the nineteenth century. Such elements have only a remote connection with algebraic number ideals, but in both cases the methodology of generalization by extension to regular- ize exceptions is the same. A feature of this theory that strikes those approaching it for the first time as rather peculiar is characteristic of much of Dcdckind’s thinking about number: a strictly finite problem is solved in terms of infinite classes. The problem for algebraic integers is that of unique decomposition; Dedckind’s solution is through the particular infinite classes of algebraic integers which he called ideals. His theory (1S72) of the real number system is based on a similar escape from the finite to the infinite by means of what he called cuts. To define "V/I, for example, Dcdckind imagined all rational numbers to be separated into two classes, say L, U\L contains all those rational numbers, and only those, whose squares arc less than 3, and U all those, and only those, whose squares arc greater than 3;L, U are said to define a ‘cut’ in the system of all real numbers, and this particular cut is * r~ said to define \3. The arithmetic of Dcdckind cuts is a map of the usual prop- erties of real numbers, such as \2 X "Vi = \2 X 3, familiar to analysts and algebraists through centuries of formal manipuia- 226 THE DEVELOPMENT OF MATHEMATICS tions. The purely formal work with irrationals produced con- sistent numerical approximations, and was sufficient for scientific applications of analysis. Dedekind aimed to provide a sound logical basis for the traditional formalism of number. The out- come of his efforts was a deeper formalism of the infinite. In 1926, the leading mathematician of his age, D. Hilbert (1862-1943, German) asserted that “The significance of the infinite in mathe- matics has not been completely clarified.” Nor had it by 1945. Further developments Dedekind’s theory of ideals was but one of several con- structed for the purpose of restoring the fundamental theorem of rational arithmetic to algebraic numbers. The other which has survived 5 is Kronecker’s theory of 1881, already mentioned in connection with complex numbers. Both Kronecker’s and Dede- kind’s theories have extensive ramifications in other departments of mathematics, and both exerted a decisive influence on the development of modern abstract algebra. A third theory has become prominent since its creation in the 1900’s by K. Hensel, in which numbers are represented by power series. This theory originated in the remark that any rational integer can be developed into a series of positive integral powers of a given prime p , with coefficients chosen from 0, 1, ... , p — 1. It may be considered as the ultimate extension of the Babylonian, Mayan, and Hindu place-systems of numeration in common arithmetic. Analogies with the theory of functions of a complex variable, also with the theory of algebraic functions of one variable and their representation on Riemann surfaces, appear to have guided this arithmetical theory in its rapid development. ‘Algebraic function’ is used here in its customary technical sense: if P(zv, z) = 0, where P is a polynomial, w is called an algebraic function of z. We shall see later that the detailed study of such functions and their integrals was a major activity of nineteenth-century mathematics. It may be of interest to indicate very briefly how the concept of arithmetical divisibility as generalized to class-inclusion by Dedekind and Kronecker became significant in departments of mathematics far distant from arithmetic. Any detailed descrip- tion soon becomes highly technical, and we can give only enough to suggest that far-reaching applications might have been anticipated from the finished form of Dedekind’s theory and the broad outline which Kronecker left of his. ARITHMETIC GENERALIZED 22 7 A familiar example from elementary analytic geometry offers the plainest hint. If C„(x, y) = 0(n = 1, 2 , . . . , ir.) are the equations of m given plane curves, then /i (*, y)Ci(x, y) + * * * -bfm(x, y)C a (x, y) = 0, in which thc/’s are functions of x, y (or constants), not identi- cally zero, is the equation of a curve passing through the points common to the m given curves. For simplicity, let all the C’s and/’s be polynomials in x, y. Then the system of all polynomials /i(.v, y)Ci(x, }') + ••• + /„(*, y)Cn(x, y), in which the C’s are held the same and the/’s arc constants or range over all poly- nomials in x, y, contains, or ‘divides,’ any particular polynomial in the system. ‘Divides’ here is as in Dedekind’s ideals or Kro- ncckcr’s modular systems. A modular system is a set M of all polynomials in s variables .Vi, . . . , x, defined by the property that if P, Pj, P 2 belong to the system, then so do Pi *r P; and QP, where Q is any poly- nomial in Xi, .... .v,. One further definition enables us to state a capital theorem of modern algebra. A basis of a modular system M is any set of polynomials Pi, P 2 , ... of M such that every polynomial of M is expressible in the form R\B\ -{- PjP; -{***"} where Pi, P;, . . . are constants or polynomials (not necessarily belonging to M). Hilbert’s basis theorem of 1890 states that every modular system has a basis consisting of a finite number of polynomials or, equivalently, a polynomial ideal has a finite basis. Anyone might be excused for doubting this theorem until he had followed its remarkably simple proof. In fact, when Hilbert applied it to prove the fundamental theorems for algebraic forms, P. Gordan (1S37-1912, German), who had previously obtained the same theorems by laborious calculations, exclaimed “This is not mathematics; it is theology!'’ There was a double-edged truth in Gordan’s protest. Hil- bert’s theorem marks a major turning point in algebra. It was the first example to attract universal attention to the modern abstract non-calculating method. Gordan’s proofs were by highly ingenious algorithms; Hilbert's attacked the structure of the systems concerned — algebraic forms and their covariants and invariants. The algorisiic method was incapable of revealing the general underlying principle of which Gordan’s theorems are 228 THE DEVELOPMENT OF MATHEMATICS but special manifestations. We shall return to this when we consider invariance. The sharper edge of Gordan’s protest was felt only in the late 1920’s. A proof in theology, it may be recalled, usually demon- strates the existence of some entity without exhibiting the entity or providing any method for doing so in a finite number of humanly performable operations. Mathematics, particularly analysis, abounds in proofs theological in this sense. To Kro- necker, all theological proofs in mathematics were anathema. He insisted that those existence proofs are invalid and therefore worthless in mathematics which do not provide a method for exhibiting, or constructing, in a finite number of humanly per- formable operations, the mathematical object whose existence is alleged to be proved. To most algebraists it is intuitive that a polynomial P(x ) with rational coefficients either is or is not rationally reducible — the product of two polynomials in x with rational coefficients. Kronecker would not admit this statement until he had devised a method for deciding in a finite number of steps whether a polynomial is actually reducible or irreducible. Since Kronecker first demanded constructive existence proofs, it has been suspected by some that the free use of ‘theo- logical’ existence proofs may lead to inconsistencies. In par- ticular, the admissibility of Hilbert’s non-constructive existence proof for his basis theorem was questioned in the 1930’s, although neither he nor the majority of working mathematicians sensed anything objectionable or dangerous in the continued application of the theorem. Without it, a vast tract of modern abstract algebra and a considerable amount of algebraic geometry would evaporate into nothing. A finitely constructive existence proof of the basis theorem had not been given up to 1945. The implied doubts in this lack are of a piece with those arising from the work of the nineteenth century on the real number system. None of these deep uncertainties deters mathematicians in their technical labors, any more than an occasional eruption discourages the vineyardists on the slopes of Etna and Vesuvius. The periodic upheavals and submersions under rivers of incan- descent lava, are indeed regarded as blessings, except by the generations who must endure them. The decomposing lava revitalizes the exhausted soil, and the grapes produce a richer wine. But it is rather unpleasant for those who must be suf- focated or incinerated in order that their successors may prosper. Much of the mathematics of the nineteenth and twentieth cen- ARITHMETIC GENERALIZED 229 tunes seems now to be significant chiefly because it may contribute to a sounder mathematical prosperity in the twenty- first century. But we have no assurance that it will. In the meantime, our generation endures or enjoys metamathematics and continues to create new mathematics. And so has it been since existence proofs svere first questioned. The general gain to 1910 What may be called the second heroic age of the theory of algebraic numbers ended in the 1870’5-1880’s in the work of Dedekind and Kroncckcr. The first great age was that of Gauss and Kummer in the 1830’s-l 840’s. The principal innovations in each of these periods naturally inspired a considerable number of technical developments. But it docs not appear that any funda- mentally new concept, or any novel approach comparable in general significance to those indicated, was suggested earlier than the third great epoch beginning in 1 910 and the I920’s. It must be recalled here that we arc interested primarily in the development of mathematical thought as a whole, rather than in the detailed exploitation of special fields. Before passing on to the third period and its relation to the general progress, we may glance back at the first two and note once more their origin, in order to see their principal residue. Perhaps the most signifi- cant contribution is the methodological approach in both periods. The theory of algebraic numbers originated in two definite problems concerning the rational integers: the laws of n - ic reciprocity, designed to yield criteria for the solvability of binomial congruences .r” ~ r mod m\ the proof or disproof of Fermat’s last theorem. A solution 5 of the first problem for « prime was given in what long remained its classic form by F. M. G. Eisenstcin (1823-1852, German) in 184*1—50, and by Kummer in 1850-61. Thus, in this direction, algebraic numbers accomplished the purpose for which they were invented. The underlying structure of the modern theory of reciprocity laws, dating from about 190S, is that of the modernized Galois theory of fields and finite groups. The second problem — Fermat's last theorem — responsible for the theory of algebraic numbers has resisted the best efforts of three generations of arithmeticians since Kummer made the first notable progress. In this direction, then, the theory has not attained its goal, although it has found much on the way. Of 230 THE DEVELOPMENT OF MATHEMATICS both problems it seems fair to say that they, as definite ends, have waned in interest, while the methods devised for their solution have steadily waxed in importance for modern mathe- matics. Algebraists, for example, who have but a slight interest in either reciprocity laws or Fermat’s theorem constantly use the machinery (fields, ideals, rings, etc.) devised in the first instance to handle these problems. The like is true of the Galois theory of equations. Galois himself made the terminal contribution so far as algebraic equa- tions are concerned, and subsequent reworkings of his initial theory have added nothing basically new to his criteria for solva- bility by radicals. Even the modernized presentation of the Galois theory, as in the streamlined model of E. Artin (Ger- many, U.S.A.), is a tribute to the mathematical creed of Galois, in its elimination of all superfluous machinery. For this modern release from algebraic calculation, the direct approach of A. E. ‘Emmy’ Noether (1882-1935, Germany, U.S.A.) in the 1920’s was primarily responsible. Much of her mathematics was in the spirit of Galois. But his methods, sharpened and generalized by his successors, have transcended the problem for which they were invented, and have rejuvenated much of living pure mathematics. It is to be noticed concerning the vital residue of the theory of algebraic numbers that it, like the Galois theory, can be traced to definite, highly special problems. Neither Galois nor the creators of the theory of algebraic number fields set out deliberately to revolutionize a mathematical technique; their comprehensive methods were invented to solve specific problems. Such appears to have been the usual path to abstractness, generality, and increased power. Some difficult problem that has appeared in the historical development of a particular sub- ject is taken as the point of departure without any conscious effort to create a comprehensive theory; repeated failures to achieve a solution by known procedures force the invention of new methods; and finally, the new methods, having been neces- sitated by a problem which appeared in the historical develop- ment, themselves pass into the main stream. Both Fermat’s last theorem and the arithmetical theory of reciprocity are but very special cases of a central problem in diophantine analysis. It is required to devise criteria to decide in a finite number of non-tentative steps whether or not a given diophantine equation is solvable. The extreme complexity of ARITHMETIC GLXERALI7.ED 231 ihe theories invented for the two special cases suggests that only insignificant progress toward a solution of the general problem is likely without the invention of radically new methods. The contribution from algebraic equations The third great epoch in the extension of arithmetic is that of the twentieth century after 1910. To anticipate, the introduc- tion of general methods into linear algebra, beginning in the first decade of the twentieth century, prepared that vast field of mathematics, first opened up by Hamilton and Grassmann in the 1840’s, for partial arithmetization in the second and third decades of the century. In 1910, E. Steinitz (1871-?, Germany), proceeding from, and partly generalizing, Kronccker’s theory (1881) of “algebraic magnitudes,” made a fundamental con- tribution to the modern theory of (commutative) fields. His work was one of the strongest impulses to the abstract algebra of the 1920’s and 1930’s, witli its accompanying generalized arith- metic. The outstanding figure in the later phase of this develop- ment is usually considered to have been Emmy Noether 7 (1882-1935, Germany) who, with her numerous pupils, laid down the broad foundations of the modern abstract theory of ideals, also a great deal more in the domain of modern algebra. The application of this work to the ‘integers’ of linear associative algebras affords the ultimate extension up to 1945 of common arithmetic. One of the main clues threading this intricate maze is the Galois theory of fields as it has developed since 1830. The Galois theory of equations itself was the concluding episode in about three centuries of effort to penetrate the arithmetical nature of the roots of algebraic equations. Accordingly we shall consider this first. It is of interest in this connection to recall an opinion expressed by Hilbert* in 1893 which still retains its force: With Gau*% Jacobi and L. Diridilct frequently and forcefully expressed their astonishment at the close connection between arithmetical questions and certain algebraic problems, in particular with the problem of cyclotomy. The baric reason for these connections is now completely disclosed. The theory of algebraic numbers and the Galois theory of equations have their common root in the theory of algebraic fields. . . . After the solution of the general cubic and quartic in the sixteenth century, there appears to have been only one con- tribution of lasting significance to the algebraic solution of 232 THE DEVELOPMENT OF MATHEMATICS equations before the late eighteenth century. E. W. Tschirn- hausen® (or Tschirnhaus, 1651-1708, German) in 1683 applied a rational substitution — reducible to a polynomial substitution to remove certain terms from a given equation. This generalized the removal of the second term from cubics and quartics by Car- dan, Vieta, and others. About a century later (1786), E. S. Bring (1736-1798, Swedish) reduced 10 the general quintic to one of its trinomial forms, x 5 + ax + b = 0, by a Tschirnhaus transformation with coefficients involving one cube root and three square roots, a result of capital importance in the trans- cendental solution of the quintic. Euler, about 1770, solved the general quartic by a method differing from that of Ferrari. This unexpected success led him to believe that the general equation is solvable by radicals. As remarked in connection with the Greek problem of trisecting an angle, it demanded originality of a high order to doubt the possibility of a solution by radicals in the general case. Were such a solution for the general quintic possible, Euler no doubt ■would have found it; for he was without a superior on the manip- ulative side of algebra. But the quintic called for a different kind of mathematics. As Abel pointed out, failure to solve the general quintic by radicals might indicate only incapacity on the part of the would-be solver; and no number of failures could be of any value as an indication whether the problem was solvable. A long stride forward was taken by Lagrange 11 in 1770-1. Instead of trying to solve the general quintic by ingenious tricks, Lagrange critically examined the extant solutions of the equa- tions of degrees 2, 3, 4 in a successful attempt to discover why the particular devices used by his predecessors had succeeded. He found that in each instance the solution is reducible to that of an equation of lower degree, whose roots are linear functions of the roots of the given equation and roots of unity. Here at last was a seemingly universal method. But on applying his reduction to the general quintic, Lagrange obtained a sextic. The degree of the resolvent equation, instead of being reduced as before, was raised. We see now that this was a strong hint of the impossibility of a solution by radicals; but Lagrange apparently missed it. He had, however, found the germ of the theory of permutation grou P s - , , In this discovery, Lagrange took the first step toward tne general theory of groups, a step of immeasurably greater signifi- cance for mathematics as a whole than a complete disposal of the ARITHMETIC GENERALIZED 233 theory of algebraic equations. Permutation groups suggested abstract finite groups. These in turn suggested infinite discon- tinuous groups, and finally the group concept entered analysis and geometry with the invention by M. S. Lie (1842-1899, Norwegian) of continuous groups in the 1870’s. The reaction upon both the discrete and the continuous divisions of mathe- matics was far reaching and profound. With invariance, closely related to the group concept, the theory of groups in the nine- teenth century transformed and unified widely separated tracts of mathematics by revealing unsuspected similarities of structure in diverse theories. This, however, belongs to the subsequent development of Lagrange’s discovery, and will be considered in the proper connections. For the moment we are concerned only with the application of groups to algebraic equations. To recall briefly the nature of what Lagrange found, let .Yi, . . . , „r„ denote the roots of the general equation of degree n. Then, if a rational function/ of a*i, . . . , x n is left unaltered by all those permutations on Xi, . . . , ,v„ that leave unaltered another rational function g of .Vi, . . . , a*„, / is a rational func- tion of g and the coefficients of the general equation. A set of permutations Si, . . . , S r on a given set of letters (as Xi, . . . , x n above) form a group in the technical sense already defined, when a product, such as SiS„ of two permuta- tions S;, Sj, is interpreted as the permutation which results when Si is applied first, and Sj is then applied to the new arrangement of .Vi, . . . , x„ generated by S,-. For example, if n = 4, and the 4 letters arc a, b, c, d , the symbol ( abed) means the permutation which takes each letter into its immediate successor, a being considered the successor of d in this cycle: a into b, b into c, c into d, d into a. The permutation {acd) takes a into c, c into d, d into a. Hence ( abed) (acd) takes a into b, b into d, d into c, and c into a. Thus ( abcd)(acd) = ( abdc ). The identical permutation /, or the ‘identity,’ takes each letter into itself or, what is the same, leaves each arrangement of the letters unaltered. The set of all possible ?:! permutations of x i} . . . , .v„ is called the symmetric group on .v t , . . . , a>.. If .Vi, . . . , x n denote the roots of an irreducible equation of degree n, the properties of the symmetric group on x t , . . . , .v„ arc the clue to necessary and sufficient conditions that the equation be solvable by radi- cals. It is impossible to go into details here, and wc must refer to any modern text on the theory' of equations or higher algebra. The fundamental concept for applications of finite groups to 234 THE DEVELOPMENT OF MATHEMATICS, algebraic equations is that of a solvable group. The meaning of this term will be explained presently. Lagrange did not explicitly recognize groups. Nevertheless, he obtained equivalents for some of the simpler properties of permutation groups. For example, one of his results, in modern terminology, states that the order of a subgroup of a finite group divides the order of the group. Normal (self-conjugate, invariant) subgroups, basic in the theory of algebraic equations and in that of group structure, were introduced by Galois, who also invented the term c group.’ Both Abel and Galois were indebted to Lagrange in their own profounder work on algebraic equations. Before Abel set himself (1824) the problem of proving the impossibility of solving by radicals the general equation of degree greater than four, an Italian physician, P. Ruffini (1765-1822), beginning in 1799, had attempted to do the same. Ruffini’s definitive effort (1813) is said by some who have examined it to be essentially the same as Wantzel’s simplification of Abel’s proof. Abel published this proof at his own expense in 1824; it was reprinted in 1826 by A. L. Crelle (1780-1855, German) in the initial volume of his great journal. Remediable defects are said by some competent algebraists to mar the final proofs of both Ruffini and Abel. But as the oversights are not fatal, it is customary to say that each of these two proved the impossibility of solving by radicals the general equation of degree greater than four. Their work was entirely independent. The unique importance of Abel’s proof is that it inspired Galois to seek a deeper source of solvability, which he found in the theorem that an algebraic equation is solvable by radicals if and only if its group, for the field of its coefficients, is solvable. We cannot enter into the technicalities of Galois’ theorem. But assuming some acquaintance with the modern theory of algebraic equations, which, after all, is well over a century old as this is written, we shall use a few of its concepts to illustrate the meaning of structure as exemplified in this capital theorem of algebra. The simple isomorphism of any two groups was defined in connection with the postulates for a group. Galois considered simply isomorphic groups as the same group which, abstractly, they are. A. Cayley 12 (1821-1895, English) in 1878 expressed this by saying that the properties of a group are defined by its multiplication table. A subgroup Hi of a group G is said to be a normal divisor of ARITHMETIC GENERALIZED 2 35 G if for every s in G, sH\ = II \S, where slh denotes the set of all products sh, h in II 1 , and similarly for II\S and the set of products hs\ equality here means that the two sets contain the same ele- ments. A subgroup of G other than G itself is called a proper subgroup. A maximal normal divisor of G is a proper normal divisor of G that is not a proper subgroup of any proper normal divisor of G. The maximal normal divisors of the group of n! permutations of the roots of the general equation of degree n appear in the criteria for solvability by radicals. To state the con- nection, we require the definition of quotient (or factor) groups. The order of a group is the number of distinct elements in the group; and Lagrange proved (1770-1) in effect that the order of a subgroup divides the order of the whole group. If Hi is a normal divisor of order m x of a group G of order n, then must n = Wi?ij ?i an integer; and it can be shown that G = Ih + silli -{-•••+ s q ^ilh, where no two of the sets II i, s x Hi, . . . , have an element in common, and the plus signs mean that all the elements of G arc separated out into these q x mutually exclusive sets. That is, the + is logical addition, as in the Boolean algebra of classes. Let A',, A' ; , . . . , A',, denote these q x sets (in any order). Then, if KiKj denotes the set of all products formed by multiplying an clement of K { by an element of A'/, it can be shown that pre- cisely mi of these products are distinct, and that these m } are all the elements of some one of the AT’s. Moreover, with multi- plication KiKj as just described, K x , A'-, . . . , A 5j form a group, called the quotient (or factor) group of G with respect to the maximal normal divisor Ih of G. This quotient group is denoted by G///i; its order is q x , and the order v.{= m x qi) of G, divided by the order (in.) of Ih, is called the index of Ih under G. Thus the index of Ih under G is here a j. Now there may be more than one proper maximal normal divisor of G. If there is, its quotient group can be formed as above, and its index under G is known. Our concern here is with all of these possibilities at each stage of the process next described. Proceeding with Ih as we did with G, we find its quotient group Ih! Ih with respect to any maximal proper normal di- visor Ih of //j. This divisor may be only the identity-group con- sisting of the single element I (the identity) of G. The process is now repeated with Ih. and so on, until it stops automatically 236 THE DEVELOPMENT OF MATHEMATICS with I. In this way, starting with G and ending with I, we get the sequence of groups G, Hi, H 2 , ... , H t ( = I), each of which (after G) is a proper maximal normal divisor of its immediate predecessor. There is also determined the sequence of quotient groups G/ Hi, HijHi, H 2 /H 2 , . . . , Hi_i/H t , and the corre- sponding indices, say qi, q 2 , ... , q t . Two final definitions, and we can state several striking consequences of this iterated process. A group having no normal divisors except itself and the identity-group I, formed of the single element I (the identity of the group), is said to be simple. If all the indices q lf q 2 , ... , q t are prime numbers, the group G is said to be solvable. Remembering that there may be several ways of proceeding at each step, we state the following conclusions. First, 13 in what- ever way we proceed, we get the same number of groups G, Hi, H 2 , . . . Hi. Second, all the factor groups displayed above arc simple. Third, 14 in whatever way we proceed, the factor groups are the same, although not necessarily in the same order, and hence similarly for the indices qi, q 2 , , q t . In a sense which need not be elaborated, these theorems of C. Jordan (1838-1922, French) (1870) and O. Holder (1859-1937, German) (1889) are a remarkable revelation of the structure of any finite discrete group. Since 1930 they have been refined and extended in what may be likened to the minute anatomy of any articulated organ- ism. 15 Recalling the capital theorem of Galois for the solvability of an algebraic equation by radicals, we see that these theorems go to the root of the matter. To anticipate slightly, it may be noted here that the Jordan-Holder theorems have themselves been structurally analyzed as phenomena of the theory of ‘lattices’ or ‘structures.’ This theory will be discussed in the next chapter. The further development of the theory of groups will be de- scribed presently. For the moment we note that the question of solvability by radicals received a conclusive answer in the theory of finite groups.\ After the impossibility of solving the general equation of degree higher thian the fourth by radicals had been proved, the next problem wa.s to find what kind of functions would suffice to solve the general ouintic. The general cubic had long been known to be solvable by circular (trigonometric) functions. The circular functions are uniform (single-valued) singly periodic functions of one variable. They are degenerate forms of the elliptic func- tions, which are uniform doubly periodic functions of one ARITHMETIC GENERALIZED 237 variable, or ‘argument’ x . If }{x) is an elliptic function, and p\, pi arc its two periods, pi/p! is necessarily imaginary, and jf(x + «j/ji + n:pi) = f(x) for all choices of the integers «j, n ?. As will be seen when we consider analysis, Abel and Jacobi in the 1820’s discovered the elliptic functions through the inversion of elliptic integrals. 16 An extensive department of the theory of elliptic functions is the problem of the division of the periods: if there is an integer n such that nx is a period, the problem of division by n is to find elliptic functions having x as argument. This problem leads to certain algebraic equations which, for « = 2, 3, 4, 3.2*, are solvable by radicals. The degree of the equation for division by any odd n is Or — l)/2. Thus for n — 5 the degree is 12. But if n is prime, the equation is obtainable in a much simpler form, being only of degree n -f* 1. We recall that Lagrange was led to a resolvent equation of degree 6 in his attempt to solve the general equation of degree 5 by radicals. The problem of division of elliptic functions for n — 5 incidentally provided the functions of a , b which reduce the trinomial form ,v s -f- ox + b — 0 of the general quintic to an identity. The trans- cendental functions necessary to solve the general equation of the fifth degree had therefore been constructed. This unexpected result was found by C. Hermitc (1 822—3 90S, French) in 1858. Ilermite was led to it by his intimate knowledge of elliptic functions. lie observed that an equation occurring in the problem of quinquisection of elliptic functions could be transformed into Bring’s form of the general quintic. Simulta- neously, Kroncckcr was nearing the name goal by another road. Kroncckcr's method differed profoundly from Hermite’s. It was closer to what Galois might have done, had he lived. In 1853 Kroncckcr net himself the task of solving a fundamental prob- lem encountered by Abel in his attack on algebraic equations: to find the most general function of x } , . . , , x n that can be a root of an algebraic equation with coefficient in a given field. lie proved that the equations arising from the theory of division of certain transcendents suffice to solve the general equations of certain degrees, and in this way obtained transcendental solu- tions of the general cubic and quartic. He then attacked the general quintic without previous reduction of the equation by a Tschirnhaus transformation to remove certain terms. His object was to find a method capable of extension to equations of any degree. Intercut in such problems flagged during the second half of 238 THE DEVELOPMENT OF MATHEMATICS the nineteenth century. So far as the general quintic is concerned F. Klein (1849-1925, German) in 1884 reviewed 16 all the labors of his predecessors, and unified them with respect to the group of rotations of a regular icosahedron about its axes of sym- metry. The earliest discussion from the standpoint of groups of the (modular) equations arising in the division of elliptic func- tions was by Galois. As a specimen of later results in this same general field, a theorem of Hilbert may be cited: the general equation of degree nine requires for its solution functions of four arguments. In summary, the chief contribution of the theory of alge- braic equations to the number concept appears to have been the characterization of the irrationalities required for explicit general solutions. The attempt to solve the general algebraic equations of degrees higher than the fourth in terms of functions con- structed from the given coefficients by a finite number of additions, subtractions, multiplications, divisions, and root ex- tractions ended in the proof by Abel and Ruffini that such solutions do not exist. But such solutions — as for the general equations of degrees 2, 3, 4 — define certain species of irrationali- ties. It therefore became necessary to seek totally different kinds of irrationalities to effect the solution of equations of degree higher than the fourth. These were found for degree 5 in the elliptic modular functions. The work of Galois and his successors showed that the nature, or explicit definition, of the roots of an algebraic equation is reflected in the structure of the group of the equation for the field of its coefficients. This group can be determined non- tentatively in a finite number of steps, although, as Galois himself emphasized, his theory is not intended to be a practical method for solving equations. But, as stated by Hilbert, the Galois theory and the theory of algebraic numbers have their common root in that of algebraic fields. The last was initiated by Galois, developed by Dedekind and Kronecker in the mid- nineteenth century, refined and extended in the late nineteenth century by Hilbert and others, and finally, in the twentieth century, given a new direction by the work of Steinitz in 1910, and in that of E. Noether and her school since 1920. Between these later developments and the work of the nine- teenth century on algebraic numbers and algebraic number fields, the elaboration of hypercomplex number systems inter- vened. This will be our concern in the following chapter. ARITHMETIC GENERALIZED 239 It is to be noted that the principal clues to the final general- ization of arithmetic have been Dedekind’s theory of alge- braic number fields, Kroneckcr’s parallel theory of what he called “algebraic magnitudes,” and the theories of general fields, groups, and rings. A ring differs from a field in that an inverse for multiplication is not postulated. The rational integers, 0, ±1, ±2, . . . are the simplest instance of a ring; the class of all rational integers is a group with respect to addition and is closed under multiplication. In a general ring, multiplication is not assumed to be commutative. Linear associative algebras are instances of rings. Changing outlooks , 1870-1920 Allusions to groups will appear frequently as v/c proceed. From 1870 to the 1920’s, groups dominated an extensive sector of mathematical thought, and were occasionally rather rashly touted as the long-sought master key to all mathematics. The first date, 1870, marks the publication of C. Jordan’s (1838-1922, French) classic Traits des substitutions cl des equations algebriques, which contained a great deal more than its modest title indicates. This is emphasized because Jordan was one of the few leading specialists in groups who made signal contributions to other departments of mathematics, including analysis. Others in the same category were Klein, Lie, Poincare, G. Frobenius (1849— 1917, German), W. Burnside (1852-1927, English), and L. E. Dickson (1 874 — , U.S.A.). But the great majority of those who labored in groups were specialists in the narrowest sense; and some of them after 1920 were content to parrot the informed but obsolete opinions of well-rounded mathematicians on the place of groups in mathematics as a whole, without acquiring the knowledge necessary to enable them to form a reasonable personal judgment. During the fifty years from 1S70 to 1920, mathematics did not stagnate; and although groups remained one of the seemingly permanent additions to mathematical thought, informed opinion after 3920 was less immoderate than it had been in the 1890’s in its claims for the domination of groups over all mathematics, it was therefore inexcusably misleading to retain in the 1930’s inflated estimates of groups that may have been valid in the 1910’s, as if mathematics had remained stationary since the death (1912) of Poincare, even when chapter and verse for 240 THE DEVELOPMENT OF MATHEMATICS the estimates in question were cited in the works of some of the greatest mathematicians of a bygone generation. As a specific instance, a prominent specialist in finite groups reproduced in 1935 the long-since-superseded dictum of Poin- care that “The theory of groups is, as it were, the whole [our italics] of Mathematics stripped of its matter and reduced to ‘pure form,’” as if this extravagance were the considered verdict of competent opinion in 1935. That it was not, will be seen in detail when we follow certain developments of geometry since 1916, to be described in connection with that other outstanding addition of the nineteenth century to all mathematical thought, the concept of invariance. Poincare’s dictum was a gross exag- geration even when it was first uttered. At best, it was an under- standable overstatement, possibly for emphasis, that deceived nobody above mathematical illiteracy. While we may continue to remember the achievements of the great creators in mathe- matics with gratitude, we do the masters of the past a left-hand honor when we perpetuate their outmoded opinions for the misguidance of oncoming generations. Some of the more conspicuous landmarks in the development of groups after Galois may be noted here, exclusive of the con- tinuous groups discussed in a later chapter. Even before Galois coined the term ‘group,’ A. L. Cauchy (1789-1857, French) made (1815) extensive investigations in what are now called permutation groups, and discovered some of the simpler basic theorems. He returned to the subject in 1844-6, and just missed the fundamental theorem (1872) of L. Sylow (1832-1918, Norwegian), proved in most texts on the theory of finite groups. Cayley (1854) stated the earliest set of postulates for a group, thereby defining groups in the accepted technical sense. This definition sank out of sight and, as will be seen in connection with continuous groups, some of the leading experts, including Lie and Klein, occasionally used the term ‘group’ for systems which are not groups in the technical sense now universal. Con- sequently, the statements of certain theorems in older work require amendment. Another set of postulates was given (1882) by H. Weber (1842-1913, German), whose Algebra (3 vols., ed. 2, 1898-9) presented a masterly synopsis of algebra as it was at the close of the nineteenth century. In passing, there is no more instructive demonstration of the change in outlook and objectives that distinguished the algebra of the early twentieth century from that of the late nineteent ARITHMETIC GENERALIZED 241 than a comparison of Weber’s classic with an advanced treatise of the 1930's. The transition from the old to the new began in 1910 with the work of Steinitz. It used to be said before 1910 that a thorough mastery of three famous classics, Weber’s Algebra , J. G. Darboux’ (1842-1917, French) Lcqons sur la theorie generals des surfaces el les applications geomelriques du calcul infinitesimal (2 vols., 1887-8; ed. 2, 1913-15), and E. Picard’s (1856-1941, French), Traite d'analyse (3 vols., 1891-6; cd. 3, 1922-7), would suffice for a liberal education in mathe- matics, and enable a competent student to begin creation in what were then topics of living interest to research workers in mathematics. Less than a third of a century sufficed to render this particular liberal education hopelessly antiquated for any- one seeking to orient himself quickly for a creative career in vital mathematics. Those who in 1940 would arrive at the front of progress were, for the most part, taking short cuts that did not exist before 1910, or even before 1920. The first decade of the twentieth century witnessed a some- what feverish activity in the postulational analysis of groups, in which American algebraists produced numerous sets of postu- lates for groups, with full discussions of complete independence. By 1910, nobody could possibly misunderstand what a group is. In another department of finite groups, also, American alge- braists were incontinently prolific: the determination of all finite groups of a given order, especially all permutation groups on a small number of letters. One of the earliest attempts at a complete census was that (1858) of T. P. Kirkman (1806-1895), an English clergyman in a muggy parish, who claimed that his methods sufficed for an exhaustive enumeration. Kirkman will appear again in connection with topology. Not a very well- known mathematician, although rather a notorious one in his own day for the perfection of his sarcasm, Kirkman appears in retrospect to have been one of the born combinatorialists in mathematical history. For various reasons, he received practi- cally no encouragement and about as much recognition. Of Americans who made notable contributions to what may be called Kirkman’s program, F. N. Cole (1861-1927) and G. A. Miller (1S63-) were among the most prolific. Groups of linear homogeneous substitutions on n variables, also groups of such substitutions with integer coefficients, in- cluding congruence groups, were studied by many after Jordan (1870) had shown their importance in several departments of 242 THE DEI'ELOPMEXT OF MATHEMATICS algebra and analysis, including hyperelliptic functions and the geometry of plane quartics. After Jordan’s own work in this division, that of E. H. Moore 17 (1862-1932), Dickson, and H. F. Blichfeldt (1874—), all of the U.S.A., from the late lS90’s to the second decade of the twentieth century, accomplished most. Interest in this specialty collapsed after about 1918, and the more imaginative algebraists turned their efforts in other directions. By 1920, the painstaking collection and detailed analysis of special groups had become a thing of the past. Should the fruits of all this devoted labor of about half a century ever be required in either pure or applied mathematics, the toilers of the future will be spared decades of some of the hardest and most thankless drudgery ever successfully carried through in the history of algebraic computation. Finite groups are an episode in modern combinatorial analysis and, as such, are as difficult to civilize as any other phenomena in that inchoate science. A brilliant exception to purely combinatorial methods ap- peared in 1896-9 in the algorithm of group-characters invented by Frobenius, and applied by him and others 18 with conspicuous success to several difficult problems in finite groups. The neces- sary computations, although often tedious, are non-tentative and non-combinatorial. They may therefore presage a more intelligently reasoned, less grubbing attack on the problem of group structure than that of the taxonomic period. With the appearance of groups in the early 1930’s in quantum mechanics, the somewhat neglected algorithm of Frobenius became of possible scientific significance, and the heavy labor of applying it in detail to the permutation groups required in physics was undertaken. So possibly science may stimulate the algebra of the future to devise more practicable methods of calculation and enumeration in finite groups than those of the heroic age of uninspired hard labor. Mathematics and society In reviewing the contribution of algebraic equations to the development of the number system, any mathematician today must be impressed by the apparent permanence of the ideas introduced by Abel (1803-1829) and Galois (1 SI 1—1 832), and the profound difference between their approach to mathematic^ and that of their predecessors including, in some respects, Gauss ARITHMETIC GENERALIZED 2-13 (1777-1855). To these young men, perhaps more than to any other two mathematicians, can be traced the pursuit of general- ity whch distinguishes the mathematics of the recent period, beginning with Gauss in 1801, from that of the middle period. They initiated for the whole of mathematics the deliberate search for inclusive methods and comprehensive theories. Their forerunners in the middle period were Descartes with his general method in geometry; Newton and Leibniz with the differential and integral calculus created to attack the mathematics of continuity by a uniform procedure; and Lagrange, with his universal method in mechanics. Their contemporary in recent mathematics was Gauss, who in his arithmetic sought to unify much of the uncorrelated work of the leading arithmeticians from Fermat to Euler, Lagrange, and Legendre. Both Abel and Galois acknowledged their indebtedness to the theory of cyclotomy created by Gauss; and although they went far beyond him in their own algebra (Abel in analysis also), it is at least conceivable that neither Abel nor Galois would have chosen the road he followed had it not been for the hints in the Gaussian theory of binomial equations. Both Abel and Galois died long before their time, Abel at the age of twenty-seven from tuberculosis induced by poverty, Galois at twenty-one of a pistol shot received in a meaningless duel. When Abel’s genius was recognized, he was subsidized by friends and the Norwegian government. By nature he was genial and optimistic. Galois spent a considerable part of his five or six productive years in a hopeless fight against the stupidities and malicious jealousy of teachers and the smug indifference of academicians. Not at first quarrelsome or perverse, he became both. Whoever, if anybody, was responsible for the colossal waste represented by these two premature deaths, it seems probable that mathematics was needlessly deprived of the natural success- sors of Gauss. \\ hat Abel and Galois might have accomplished in a normal lifetime cannot be even conjectured. That it would have been much and of the highest quality seems probable. Early maturity and sustained productivity are the rule, not the excep- tion, for the greatest mathematicians. It may be true that the most original ideas come early; but it takes time to work them out. Gauss spent about fifty years developing the inspira- tions that came to him (this is substantially his own descrip- tion) before he was twenty-one, and even with half a century of CHAPTER 11 Emergence of Structural Analysis With the material already described as a background, we shall now observe in some detail the trend toward ever greater generality and more refined abstraction which distinguished much mathematics of the recent period from nearly all that preceded 1S-10. Structure, in a sense to be noted and described, was the final outcome of this accelerated progression from the particular to the general. The entire movement may be seen in geometry as clearly as in algebra and arithmetic, and will be remarked in that connection in later chapters. The course fol- lowed here continues that already indicated merely for con- venience. It is typical of that for all divisions. Wc saw that Gauss in 1851 invented his complex integers .Vj + :.%•» to solve a specific problem in rational arithmetic. His .v» -f t.V; written as a number-couple (xi, x : ) suggested hyper- complex numbers (.\*j, . . . . ay) with n coordinates -Vj, . . . ,.r n ; and it was natural to ask whether any of these extended num- bers, with real or ordinary complex coordinates, might be useful in rational arithmetic. More generally, how may ‘integers’ be defined in a system of hypcrcomplcx numbers, and what is their ‘arithmetic’? Before cither problem could be attacked, or even formulated precisely, the algebra of hypcrcomplcx number systems had to be developed. But this in its turn was not a definite problem. Once the algebraic problem had been made precise, its solution followed rapidly. 1 here appear to have been three principal phases after the problem was first posed by the invention of quaternions. 246 THE DEVELOPMENT OF MATHEMATICS Three phases in linear algebra The first phase was represented by such work as that of B. Peirce in 1870, which sought to find and exhibit all the linear associative algebras in a given (finite) number of fundamental units. 1 The second phase, merging into the third, began in the first decade of the twentieth century and continued to about 1920. In this period, general theorems applicable to all linear associative algebras were the objective. The third phase was distinguished for its restatement in abstract form of much that was already known, and the intro- duction of arithmetical concepts, such as ideals and valuations, into the resulting abstract algebra. The outcome 2 was an exten- sive and intricate theory assignable to either algebra or arith- metic, according to taste. The algebraic number rings and fields long familiar in the theories of algebraic equations and algebraic numbers; the ideals and dual groups of Dedekind; the relative fields of Hilbert; the modular systems of Kronecker; and the Galois theory of fields, all contributed to the final abstract theory. The finished product exhibits the broad outlines of the theories from which it evolved as but different, particularized aspects of a unified whole, like the varying projections of an intricate geometrical configuration on a moving plane. In addi- tion, the abstract theory gives a wealth of results not obtainable from its classical instances. The abstract method The entire development required about a century. Its progress is typical of the evolution of any major mathematical discipline of the recent period; first the discovery of isolated phenomena; then the recognition of certain features common to all; next the search for further instances, their detailed calcu- lation and classification; then the emergence of general principles making further calculations, unless needed for some definite application, superfluous; and last, the formulation of postulates crystallizing in abstract form the structure of the system investigated. The detailed elaboration of the abstract system implicit in the postulates then proceeds undistracted by what may be adventitious circumstances in any special instance. Incidentally, this is the reason that our extremely practical decadic Hindu-Arabic numerals are a positive detriment, except EMERGENCE OF STRUCTURAL ANALYSIS 247 for numerical checks, in investigating the properties of numbers. The p-adic and g-adic numbers of Hensel are closer to arithmetic. The full import of the abstract formulation appears only when it is taken as the point of departure for the deliberate creation of netv mathematics. Certain postulates in the original set are suppressed or contradicted, and the consequences of the modified set are then worked out as were those of the original. For example, in a field as first defined, multiplication is commuta- tive. This raises the question whether it is possible to construct consistent ‘algebras’ subject to all the postulates of common algebra except the commutativity of multiplication. Again, in common algebra (a commutative field), if a 0, and ab = ac, it follows by division that b — c. But division by a presupposes that a has an inverse with respect to multiplication. Division is not defined in a ring; nevertheless there arc rings in which, if e ~ 0 and ab — ac , then b = c. Hence the existence of inverses as in a field is not postulated, but is replaced by the weaker condition just stated taken as a postulate. The result is a type of algebra more general than a field, in that it is based on some but not all of the field postulates or their consequences. While viewing the abstract method we need not let our sincere admiration for its undeniable beauties betray us to the fate of Narcissus. It is just possible that our descendants may record that we perished of hunger while staring at the seductive reflection of our own superficial ideas — “What thou scest, What there thou scest, fair creature, is thyself — ”, 3 or that we fell in and were drowned in something we never sus- pected just below the entrancing surface. To illuminate this heresy with an even more heretical example, what reason is there for supposing that because Dedekind's ideals did what was required for algebraic number rings, something very much like them should be introduced into other rings? Is it even evident that the arithmetic of a non-commutative ring should be based on ideals at all ? Or that the usual definition of integral elements of a ring by close analogy with the like — by means of the rank equation — for an algebraic number ring is the most promising clue: The obvious retort is to demand something better of the doubter. Admitting the justice of this, we may nevertheless consider the other possibility. 248 THE DEVELOPMENT OF MATHEMATICS The root of these doubts seems to be the unimaginative lack of a clearly recognized objective. If the aim is merely to create new theories which many find intensely interesting and even beautiful, then the abstract method keeps on reaching its goal. In this respect it somewhat resembles Stephen Leacock’s hero who leapt on his horse and dashed furiously off in all directions. But the taste of another generation may find our abstractions boring and our beauties vapid. They will have a hard way to travel before coming upon something different. The tangled masses of our theories will impede their every step. If they are to progress, their only possible road will circumvent our work al- most entirely. The like happened once before, when Descartes walked clear round the synthetic labors of Euclid and Apollonius. To the skeptically inclined, viewing the vast accumulations in abstract geometry, abstract algebra, and abstract analysis of the twentieth century, another Descartes seems about due. Unless he arrives within the next two thousand years, no two mathe- maticians in the world twenty centuries hence will understand each other’s words. In the meantime we may appreciate the tremendous accomplishments of the modern abstract method and trace the main steps by which it finally arrived since Hilbert gave it the strongest impulse since Euclid in his geometry of 1899. The first phase in the development of abstract algebra, that of calculation and tabulation, is on the same scientific level as systematic botany. All sciences of the past seem to have been condemned to creep through this Linnaean stage of development. If only Linnaeus would stay as dead in mathematics as he appears to be in biology, mathematics might be far leaner and more virile than it is. But the irrepressible botanist keeps on rising from the dead; it is impossible to keep him down. Idle the vanguard of mathematical progress is advancing to genera, a host of straggling camp followers busies itself with the collection and classification of trivial or discarded subspecies. It rvas so in the theories of algebraic in variants and finite groups. An exas- perating instance from recent analysis is the introduction of tv.o new technical terms to distinguish a: > 0, x S 0. After enough classification has been done to indicate some inclusive character- istic, there would seem to be no point in collecting further specimens unless they are to be used. Linear algebra fortunate!) escaped the intensest fury of the taxonomists in its rapid passage to the second phase. EMERGENCE OF STRUCTURAL ANALYSIS 249 Totvard structure in algebra We shall select a few typical episodes from the history of linear algebra* to illustrate the general trend since about 1870, in nearly all mathematics, from the detailed elaboration of special theories to the investigation of interrelations between the theories themselves. Technicalities are unavoidable, "but they may be ignored by those unfamiliar with the subject; the important items are not the facts but the relations between them. Even without technical knowledge, it is possible to appre- ciate distinctions in scope and generality between different theorems. The technicalities, however, are of interest on their own account. Some of them are landmarks in their own province. Only what appear to have been the principal steps leading from the enumerative stage of hypercomplex number systems to the abstract theory of all such ‘algebras’ will be considered here. The work of B. Peirce (1S70) aimed at principles for the ex- haustive tabulation of linear associative algebras in a given finite number of fundamental units, with real or complex number coefficients. His methods were adequate, and his partial failure to find all the algebras he sought in less than seven units was due to mere oversights. Peirce’s problem was equivalent to that of exhibiting all sets of n linearly independent symbols (the basal, or fundamental, units) €\ forming closed systems under associative multiplication, on the assumption that any product e T e, is a linear function of Ci, , e n . For n given, the problem can be solved brutally by actually constructing all possible multipli- cation tables, applying the associativity condition — {e T e,)e t , and retaining only those which actually dose. The labor of such an undertaking quickly becomes prohibitive with increasing n, and Peirce proceeded otherwise. Two of his guiding principles depended on the presence or absence of nilpotent and idempotent units: if there is a positive integer r > 1 such that c T — 0, Peirce called c nilpotent; and similarly for e- — c and idempotent. These pervasive concepts of linear algebra were the foundation of Peirce’s classification. His work was continued bv his son, C. S. Peirce (1859-1914, U.S.A.), who also made outstanding contributions to mathematical logic and is said to have invented the peculiarly Yankee philosophy known as pragmatism. 250 THE DEVELOPMENT OP MATHEMATICS In an appendix to his father’s memoir, C. S. Peirce proved a famous theorem, 5 which we restate in its customary form: The only linear associative algebras in which the coordinates are real numbers, and in which a product vanishes only if one factor is zero, are the field of real numbers, the field of ordinary com- plex numbers, and the algebra of quaternions with real coeffi- cients. The advance here beyond tabulation is evident. The theorem also indicates one possible kind of answer to the question asked by Gauss. It suggests that quaternions might have a useful arithmetic, as they are so closely related to ordinary complex numbers. Extensive arithmetics of ordinary quaternions were constructed by R. Lipschitz (1832-1903, German) in 1886, and by A. Hurwitz in 1896. Dickson in 1922 simplified these arithmetics. The historical order here is that of increasing simplicity. From the late 1870’s through the 1890’s, linear algebra took several new directions which looked extremely promising at the time, but which do not appear to have influenced the main advance significantly. Thus G. Frobenius (1849-1917, German) in 1877 developed an interesting connection between hypercom- plex number systems and bilinear forms. This was followed in 1884 by H. Poincare’s discovery of a similar connection with the continuous groups due to M. S. Lie, (1842-1899, Norwegian). Lie’s outstanding contributions to nineteenth-century mathe- matics will be noted in connection with invariance. Poincare replaced the problem of classification by an approachable equivalent for a wide class of linear algebras: to find all con- tinuous groups of linear substitutions whose coefficients are linear functions of n arbitrary parameters. This line of attack was developed with considerable succesjs by G. W. Scheffers (1866-, German), a scientific legatee of Die, who in 1891 under- took to show that Lie’s theory of (finite) continuous groups contains the theory of hypercomplex / number systems. The theory of such groups therefore afforded principles for the classification of linear associative algebras. In the 1890’s Lie’s theory was perhaps more assiduously cul- tivated than it was to be for a generation, when it was revived in a renovated differential geometry. Reflecting this widespread interest, E. Cartan (1869-, French) in. 1898 applied the Lie theory to obtain a classification of hypercomplex number sys- tems which followed ‘naturally’ from that theory. But sue apparently promising leads were practically abandoned short v EMERGENCE OF STRUCTURAL ANALYSIS 251 after 1900, and the main advance proceeded in another direc- tion, beginning in 1907 with the work of J. H. M. Wedderburn (1882-, Scotch, U.S.A.). Frobenius and Cartan had obtained numerous special results for hypercomplex number systems in which the coordinates are rational numbers, whereas in a general theory the coordinates should be elements of nothing more restricted than an abstract field. The postulates for such a situation were formulated in 1905 by Dickson. As much of Cartan’s development depended on the characteristic equation of a linear associative algebra, it was not always extensible to the general case. In any event, algebra took a new turn in 1907, heading directly toward a theory of structure. The theorem of Frobenius and Peirce on quaternions sug- gested the search for all linear associative® algebras satisfying certain preassigned conditions. The most important of these are the division algebras, in which, if a(y£ 0) and b are any elements of the algebra, each of the equations ax = b, ya = b has a unique solution. Two of the earlier results on division algebras may be recalled, to provide concrete examples of structure. Galois initiated the study of fields containing only a finite number of distinct elements. It was proved in 1893 by E. Ii. Moore (1862-1932, U.S.A.) that every finite commutative field is of the type considered by Galois; that such a field is uniquely determined by a pair of positive integers p, n, of which p is prime; and that the corresponding field contains p n distinct elements. This theorem exhibits one of the many variants of structure: an exhaustive characterization is prescribed for all (commutative) fields containing only a finite number of distinct elements. Another variant appears in Wcdderburn’s theorem of 1905, that if the coordinates of a linear associative division algebra arc elements of a finite field, then necessarily multiplica- tion in the algebra is commutative. As will be seen presently, division algebras play a dominant part in the theory of algebraic structure. The determination of all division algebras, or the discovery of comprehensive classes of such algebras, thus became a central problem in the theory. Dickson in 191-1 constructed division algebras in r. fundamental units with coefficients in any field F. At this point we may repeat that we arc primarily interested here in the emergence of the theory of structure, particularly as exemplified in linear algebra. To bring out the essentials, it will be necessary next to state a rather formidable-looking theorem 252 THE DEVELOPMENT OF MATHEMATICS containing several technical terms whose meaning has not been explained. Those already acquainted with the subject will rec- ognize the statement as one of the fundamental theorems' (Wedderburn’s, 1907) on the structure of algebras. Those seeing it for the first time may substitute letters S, X, Y, . . . for the technical terms ‘sum/ ‘semi-simple/ ‘direct sum/ . . . , as we shall do in a moment, and attend solely to the construction of the sentences constituting the theorem. The structural charac- ter, which is our concern here, is then evident. As a mere verbal convenience, a linear associative algebra whose coordinates are in a field F is said to be over F. The theorem states that: (1) Any linear associative algebra over a field F is the sum of a semi-simple algebra and a nilpotent invariant subalgebra, each over F ; (2) A semi-simple algebra over F is either simple or the direct sum of simple algebras over F. (3) Any simple algebra over F is the direct product of a di- vision algebra and a simple matric algebra, each over F, includ- ing the possibility that the modulus is the only unit of one factor. 8 Eliminating the technicalities, irrelevant for our purpose, we restate this with S, X. Y, . . . . (1) Any linear associative algebra over a field F is the S of an X-algebra and a E-algebra, each over F; (2) An X’-algebra over F is either a Z-algebra or the DS of Z-algebras over F: (3) Any Z-algebra over F is the DP of a W- algebra and a 17-algebra, each over F. The theorem exhibits the structure of any linear associative algebra over any (commutative) field F, in terms of three kinds of operations, S, DS, DP, and five species of algebras X, Y, Z, TV, U. Thus any linear associative algebra over any field F may be dissected into algebras of the five kinds specified, and always in the same way, namely, by S, DS, DP. Without further elaboration, this is what is meant by saying that all linear associative algebras over any field F have the same structure with respect to certain specified kinds of subalgebras. Attention may therefore be confined to the five specified kinds of sub- algebras. One of these, W, comprises the division algebras. The radical distinction between general structure theorems of this kind and the cataloguing type of algebra which preceded it is obvious. The shift of objective is typical of modem abstract EMERGENCE OF STRUCTURAL ANALYSIS 253 mathematics. Specimens are no longer prized for their own curious sake as they were in the nineteenth century. It is as if some industrious company of fossil collectors who had never heard of Darwin were suddenly enlightened by an evolutionist. Their interesting but somewhat meaningless collections would simplify themselves in an unsuspected coherence. Toward abstraction in analysis and geometry Three further advances of 1906-20 toward abstractness and generality may be mentioned here, as they are connected with the real numbers. None originated in algebra; yet at least one — J. Kiirschak’s (1864—1933, Hungary) — was to suggest far-reach- ing consequences in both algebra and arithmetic. All were in the direction of a theory of valuation generalizing that of real numbers and ordinary complex numbers. With any ordinary complex number x + iy, written as a Hamiltonian number-couple (x, y), is associated the unique real number |(.x, y)|, the ‘absolute value’ of ( x , y), which is the posi- tive square root of x~ + y~. But if x, y arc elements of an ab- stract field F, x : -f- y 2 is not a real number. To distinguish the zero clement of F from the zero, 0, of the real number field, we shall write it O’. Extending the properties of absolute values for the field of ordinary complex numbers to the elements O', x, y, . . . of an abstract field, F, Kiirscbak in 1913 associated with any clement z of F a unique real number, its ‘absolute value,’ denoted by jd, subject to the postulates [O'j = 0; ].vj > 0 if x ^ 0‘; i-v.j = Jcj \tt\i\z + tej S H + pi, for any z, tv in F. The last is sometimes called the triangular inequality. It is the analogue of the theorem in plane Euclidean geometry that any side of a triangle is less than, or equal to (when the vertices are collincar), the sum of the other two sides. If the ‘distance’ be- tween any two elements X, y be defined by |.v — yj, the postulates for these absolute values reproduce the properties usually associated with the concept of distance. They were so defined in 1906 by (R.) M. Frcchct (1878-, French) in his thesis for the doctorate at Paris. This work is one of the sources of modern gen- eral or abstract analysis, the theory of abstract spaces, and to- pology (all to be noted in other connections). A further advance in this direction was made by S. Banach (1892-1941, Polish) in 1920, who removed the restriction that the elements x, y, . . . be in a field; his .v, y, . . . are elements of any class whatever. 254 THE DEVELOPMENT OF MATHEMATICS It might be thought that nothing not already known could issue from such faithful copies of the simplest properties of real and complex numbers. To substantiate the contrary, we need cite only one rather unexpected outcome of this process of ab- straction. It has been found that much of the analysis based on the real or complex numbers has its image in general analysis. It is not necessary to assume that x, y, . . . are real or complex numbers to obtain many results which formerly were supposed to be consequences of that assumption. In Frechet’s analysis, for example, limit points and convergent series are definable, and the theorems on convergence for this abstract analysis are applicable to the special case of the analysis in which the basic elements are real or complex numbers. Again, it might be imagined that for the rational numbers r, . . . the only possible frjj is the familiar \r\. But it was shown (1918) by O. Ostrowski (1893-, Russian) that there are in fact precisely two distinct types. Thus, in this instance at least, abstraction led to some- thing new and unexpected. A terminus in arithmetic Gauss’ question concerning the possible utility of hyper- complex numbers in the higher arithmetic is of peculiar interest, both historically and mathematically. We have seen that Gauss was acquainted with quaternions, and we have referred to applications of quaternions to the classical theory of numbers. It is unlikely that Gauss had developed quaternions far enough to suspect that they have interesting arithmetical properties. Nevertheless, he asked 9 “whether the relations between things, which furnish a manifold of more than two dimensions, may not also furnish permissible kinds of magnitudes [numbers] in general arithmetic?” There has been much speculation as to what Gauss con- sidered 'permissible.’ Each guess generated its own answer in an enumeration of all algebras satisfying the ‘permissible’ condi- tion. We shall report only one of several closely similar answers, as the direction taken by general arithmetic since the time of Gauss certainly was not foreseen by him. Weierstrass is said to have proved the following theorem in his lectures of 1863; at any rate he published it, with several more of a like kind, in 1884. The only hypercomplex number systems with real coordinates, in which a product vanishes only if at least one of its factors does, and in which multiplication is EMERGENCE OF STRUCTURAL ANALYSIS 255 commutative, are the algebra with one fundamental unit c such that e- = e, and the two-unit system of ordinary complex num- bers. If Gauss did not permit divisors of zero, and if he insisted on commutative multiplication, this answered his question. The algebra with e- — c is of little interest; the other gave Gauss his own arithmetic of complex integers a -f- hi, a , b real integers. In short, he himself had reached the end of the road he may have imagined but did not explicitly indicate. Further progress was in other directions, starting in the work of Dedekind on algebraic number fields and ideals, and in Kronccker’s theory of modular systems as developed by himself and others, notably E. Lasker (1S6S-1941, German, former world chess champion) in 1905, J. IConig (1849-1913, Hungary) in 1903, and F. S. Macaulay (1862-1937, English) in 1916. We pass on to a short enumeration of what appear to have been the principal steps toward these vast developments of modem algebra and arithmetic. Newer directions From the great mass of work that has been done since 1900 on the arithmetization of algebra — or vice versa — we shall select only three items, to indicate the trend toward abstractness and the analysis of structure. In his algebraic theory of fields (1910), E. Stcinitz sought all possible types of fields and the relations between them. Pro- ceeding from the simplest types, explicitly defined, he extended these by cither algebraic or transcendental adjunctions. In al- gebraic adjunctions, Stcinitz followed the method of Cauchy as exploited by Ivroneckcr, described in an earlier chapter. The con- cepts of characteristic, prime field, complete field, and others now familiar in modern texts on higher algebra were fully treated in this profound reworking and extension of Kronecker’s theory of algebraic magnitudes. The final outcome may be roughly de- scribed as an analysis of the structure of fields with respect to their possible subficlds and supcrfields. The next item, dating from about 1920, marks a distinct ad- vance. It is represented by a host of vigorous workers who, in the twentieth century, undertook to do for an abstract ring what Dedekind had done for any ring of algebraic numbers, and to ex- tend the Galois theory to abstract fields. Thus the Dedckind theory of ideals was abstracted and generalized, as was also the Galois theory. The first of these may properly be assigned to 256 THE DEVELOPMENT OF MATHEMATICS arithmetic, as one of the chief objectives is the discovery, for any ring, of unique decomposition theorems analogous to the funda- mental theorem of arithmetic, or to the unique representation of a Dedekind ideal as a product of prime ideals. It was not to be expected that there would be a single type of decomposition obviously preferable above all others for a general ring; nor was it reasonable to suppose that rational arithmetic or the theory of algebraic numbers would be translatable into the new domain with only minor modifications. Only the arithmetic of rings with commutative multiplication had been discussed with anything approaching completeness up to 1945. In spite of radical differences between the arithmetic of commutative rings (usually without divisors of zero, the so- called domains of integrity) and that of algebraic numbers, the theory of Dedekind ideals proved a valuable clue. For example, in the Dedekind theory, an ideal has a finite basis; that is, any number of the ideal is representable as nibi -f • • * -f n,b r) where hi, . . . , b r are fixed integers of the algebraic number field concerned, and n-i, fir range independently over all integers of the field. But this fact does not directly suggest a profitable generalization to rings. However, it is equivalent to the theorem that any sequence A\, Ai, A 3 , . . . of ideals which is such that is a proper divisor of Aj, for/ = 1, 2, ... , ends after a finite number of terms. This ‘chain theorem,’ valid in Dedekind’s theory, is generalizable. Two basic but rather inconspicuous-looking items of the classical theory of algebraic number ideals passed unchanged into the abstract theory, the G.C.D. (‘greatest’ common divisor) and L.C.M. (‘least’ common multiple). Although at the first glance these are mere details, experience has shown that they are the framework of much algebraic structure and that, when their simplest properties are restated abstractly as postulates, the resulting system unifies widely separated and apparently dis- tinct theories of algebra and arithmetic. They lead, in fact, to what seemed the most promising theory of algebraic-arithmetic structure devised up to 1945. We shall therefore describe their properties in some detail. The relevant phenomena appeared first in mathematical logic, specifically in the algebra of classes, now called Boolean algebra after its founder, G. Boole (1815-1864, English). If the letters A, B, C, . . . denote any whatever classes, and if the symbol > is read ‘includes,’ or ‘contains,’ it is true that from EMERGENCE OF STRUCTURAL ANALYSIS 257 A > B and B > C follows A > C. The symbol < is read ‘is included in,’ or ‘is contained in.’ If A, B are any two classes, their ‘intersection,’ denoted by [A, 2?j, is the most inclusive — largest — class whose members arc in both A and B. For example, if A is the class of all animals with red hair on their heads, and B is the class of all girls, [A, B] is the class of all red-headed girls. If A is as before and B is the class of all vegetables, [A, B] is the null class — the class with no members. Again, if A , B are any classes, their ‘union,’ denoted by ( A , B), is the least inclusive class whose members are in A, or in B, or in both. In the second of the above examples, (A, B) is the smallest collection each of whose members is either an animal with red hair on its head, or a vegetable. We may restate these definitions as follows. If A, B are any two classes, they have a unique most inclusive common subclass, \A , B], and a unique least inclusive common superclass, {A, B). By definition, A is equal to B , -written A — B, if, and only if, A > B and B > A-, and by convention, A > A. It is now a simple exercise in language to verify the following statements concerning any set (or class) <3 of classes A , 2?, C, D, D\, . . . , M, Mi, . . . (1) If A, B, C are any three members of ©, such thaty/ > B and B > C, then A > C. (2) If A, B are any members of ©, there is a member of <3, say D, such that D g A, D £ B\ and such that, if also g. A, Di g B, then D x g D. There is also a member of ©, say M, such that M S A, M § B ; and such that, if also Mj ^ A, Mi Si B, then M\ § M. The assertions in (2) arc true when D is the intersection \A, 2?] of A , B, and 21/ is their union ( A , B). Reading ( A , ( B , C )) as the union of A and ( B , C), and [A, [2?, C]] as the intersection of A and [B, C], we have the following theorems as immediate consequences of (1), (2): [A, B), {A, B) are uniquely defined; [A, B] — [B, A], (A, B) = (2?, A)- [A, A] — A, {A, A) « A; [A, [2?, CJ] = \[A, 2?], C], (A, (2?, C)) = ((v/, 2?), C); ( A , [//, 2?]) = A, [A, (A, B)] = A. It may be left to the reader’s ingenuity to decide whether or not the next is true for classes, (3) If A < C < (A, B ), then C = (A, \B, q). It is important for our purpose to verify that (1), (2), and the simple theorems above arc satisfied for the following wholly 258 THE DEVELOPMENT OF MATHEMATICS different interpretations of A, B, C, . . . , >,=,<; A, B C, . . . is any class of positive rational integers; ‘ = ’ is equality as in common arithmetic; ‘ >’ means ‘divides’; ‘ <’ means ‘is divisible by’; [A, B] is theL.C.M. of A , B, and ( A , B) their G.C.D. Recalling that ‘divides’ in the theory of Dedekind ideals means ‘contains’ or ‘includes,’ as in inclusion for classes, we see why the interpretation in terms of classes should be relevant for the theory of ideals. We now empty A, B, C, <, [A, B], ( A , B) of all interpretation and of all meaning, and take (1), (2) as postulates defining the meaningless marks A, ...,>, <, etc. Denoting the D , M in (2) by [A, B], and (A, B) respectively — a mere convenience of notation — we can deduce from our postulates the same theorems as before. The abstract system so defined is not vacuous; for we have actually exhibited two of its instances. One would have sufficed. There are many more. The abstract system defined by (1), (2) has been called by various names, including ‘structure’ and ‘lattice.’ The second is to be preferred here, to avoid confusion with ‘structure’ as previously defined in mathematical logic. Here we reach the ultimate in arithmetized algebra or alge- braized arithmetic up to 1945. It may turn out to be a very bad guess but, as this is written, many of the younger generation of algebraists and arithmeticians believe that in this theory of lattices they have at last unified a welter of theories inherited from their prolific predecessors. The theory has been most vigorously developed in the United States. To vary the historical monotony of dwelling almost exclusively on the dead, we may mention the names of two of the most active contributors to this rapidly expanding domain of abstract mathematics, G. Birk- hoff 10 and O. Ore. The assertion (3), which was abandoned to the ingenuity of the reader, is not a consequence of (1), (2) in the abstract theory. When (3) is included with (1), (2) among the postulates, the resulting system defines a special type of lattice which is named after Dedekind, because he was the first (1897) to investigate such systems. That he did so, is but another instance of his penetrating and prophetic genius. His work passed practically unnoticed for a third of a century, when its significance was realized in the theory of lattices. _ . , To leave this comprehensive theory of lattices here with no indication of its scope would do it but scant justice. A i ery ne EMERGENCE OF STRUCTURAL ANALYSIS 259 quotation 11 from one of the creators of the theory contains the meat of the matter. “In the discussion of the structure of alge- braic domains, one is not primarily interested in the elements of these domains, but in the relation of certain distinguished sub- domains, like invariant [normal] subgroups in groups, 15 ideals in rings, and characteristic moduli in modular systems. For all of these are defined the two operations of union and intersection, satisfying the ordinary axioms” — the postulates (1), (2). The rapid expansion of this theory of structures or lattices, after a quiescence of about a third of a century following Dedckind’s introduction of dual groups, is typical of much in the recent development of mathematics. Among other common features is the apparently inevitable slowness with which a basic simplicity underlying a multitude of diversities finally emerges. Usually the unifying concept implicit in all of its different manifestations is almost disconcertingly obtdous once it is perceived. But there is as yet no recognized technique for perceiving the obvious or for not confusing significance with triviality, and each instance, it seems, must wait its own more or less random occasion. Nor are mathematicians always reliable prophets of what mathematics is to retain its vital interest or acquire a new importance. An example which deserves to become a historical classic is the sudden rise to popularity of the tensor calculus which, until the relativists adopted it, was the neglected waif of the mathematicians. In all such revaluations it is easy after the fact to sec that they might have occurred much earlier than they did. Yet nobody can foresee where the next is to happen. Mere neglect, however, is not necessarily an assurance of immortality to any who may be overanxious about their reputations or the permanence of their work. In the matter of lattices or structures, one of the deter- mining characteristics of the theory might have been anti- cipated — but was not — in 1854 when Boole published his Laws of thought. The irrelevance of the nature of the individual ele- ments of the classes whose unions and intersections give Boolean algebra its distinctive character was recognized by Boole himself. Apparently without fully realizing what he had done, and what now seems so plain, Boole had taken the first and the decisive step toward the abstract algebra and some of the geometry of the 1930’s-1940's. In these newer developments the primary interest is not in the elements of certain domains, but in the inclusions, the intersections, and the unions of distinguished 260 THE DEVELOPMENT OF MATHEMATICS subdomains formed of classes of elements of the original domain. Sets of elements rather than the elements themselves become the basic data, and the original level of abstraction rises to the next above it in what experience has shown to be a natural hierarchy of abstractions. The Boolean algebra of classes attracted but little attention from mathematicians during the nineteenth century, and the hints it might have offered projective geometers and algebraists passed unnoticed. If Boole had assimilated the controversy of the 1820’s over the validity of the geometric principle of duality, he might well have anticipated the algebraic interpretation of projective geometry of the 1930’s. The controversy will be noted in a later chapter; for the moment it suffices to state that the ‘real space’ some of the contestants imagined they were discussing evaporated. We shall return to this presently. Another clue that might have been noticed appeared in the various decomposition theorems of arithmetic and algebra. Decomposition, as in unique factorization in rational arithmetic and the theory of algebraic integers, reduces a given system to a system of simpler parts, the parts being combined according to prescribed rules to produce the elements of the given system. The ‘simpler parts’ in rational arithmetic are the rational primes; in algebraic numbers,, the prime ideals; and in both, the rules are those of multiplication as in an abelian group. The decom- position may not be unique, as for instance the basis of an abelian group. Further suggestive examples of decomposition were the Jordan-Holder theorem, described earlier, for finite groups, Wedderburn’s theorems on the structure of linear algebras, and A. E. Noether’s determination of all commutative rings in which there is unique prime ideal factorization. For the Jordan-Holder theorem it is the subsystem of normal (invariant, self-conjugate) subgroups of a given finite group with respect to which the decomposition is effected that matter; the elements of the group itself are of only minor importance. For algebraic integers the distinguished subsets are the ideals, and so on; in each instance the original elements are subsidiary. With these and other examples before them, it gradually became evident to algebraists that some common characteristic must be the ulti- mate source of at least a part of the phenomena of decomposition in the several theories. Axiomatic formulations reveal the com- mon characteristic to be an underlying lattice. In his work of 1900 on the dual groups generated by three moduli, Dedekind EMERGENCE OF STRUCTURAL ANALYSIS 261 had noticed the affiliation with what is now called Boolean alge- bra. Once the unifying feature was recognized, the natural next step was to develop the algebra of lattices as an independent theory on its own merits. The outcome was an abstract theory of structures or lattices. Abstraction for its own sake may prove fruitful or barren. If it also suggests new theorems or expresses what is already known more simply and more clearly, abstraction rises to the level of creative mathematics. As the theory of lattices advanced, it unified and simplified much of existing algebra and the funda- mental parts of certain other disciplines; it also aided in the discovery of new results. It was not to be expected that the general lattice by itself would clarify all the phenomena of de- composition, and as the theory progressed, specialized lattices of various types were defined to accommodate particular theories. One significant clue was the invariance of the chain length — the number of factors of composition — in the Jordan-Holder theorem for finite groups. Other equally patent clues might have been followed initially. But all this is in retrospect, and it will be in closer conformity with time if we merely mention some of the items which the theory of lattices illuminated. For further details we may refer to G. Birkhoff, Lattice theory, 1940. Boolean algebra, the historical source of lattice theory, found its natural place in the theory as a special type of lattice (complemented, distributive). Distributive lattices may be realized by rings of sets, an observation — in other terminologies — which goes back substantially to Euler. From this follows, on appropriate specialization, the representation theory of Boolean algebras by fields of sets. Partly suggested by the last, there arc applications of lattice algebra to the classic theories of sets and measure. In a later chapter we shall note the wide generalizations of the geometric ‘space’ of the nineteenth cen- tury which evolved into the twentieth-century geometry' of abstract space. Among these generalizations, the function spaces of the twentieth century' readily accommodated them- selves to abstraction. L. Kantorovich (Russian) essentially' defined (1957) a partially ordered linear space as a (real) linear space having non-negative elements /, symbolized / > 0, subject to the following three postulates: If / > 0 and X > 0, then V > 0; if / > 0 and — / > 0, then / = 0; if / > 0 and g > 0, then / -r £ > 0. It was shown by G. Birkhoff in 1940 that 262 THE DEVELOPMENT OF MATHEMATICS every function space then known forms a lattice with respect to this partial ordering. Vector lattices were also defined and shown to be decomposable into their suitably defined positive and negative components. In connection with abstractions of the absolute values of real and complex numbers, we shall note in a later chapter the type of abstract space named for the Polish mathematician S. Banach. Banach lattices were defined as vector lattices with a suitably specialized norm (generalized absolute value), and it was shown that all the examples given by Banach of his space are such lattices. As a last result in this direction, the decompositions of partially ordered function spaces when characterized abstractly yield components forming a Boolean algebra. From its historical origin, lattice theory might have been expected to have applications to mathematical logic and the mathematical theory of probability, and such was found to be the fact. A curious application (1936) provided a model of quantum mechanics. We recall that the concept of observables is central in this mechanics, also that the indeterminacy prin- ciple imports probability into the metaphysics of all physical observation. A basically different application (1944) of logic and probability to the quantum theory will be noted in the concluding chapter. A revealing application of lattices was the restatement of projective and affine geometries by K. Menger in 1928, Bemer- kungen zu Grundlagenfragen IV, as instances of what was later to be called lattice algebra. Menger considered a system of abstract elements for which two associative, commutative operations, denoted by +, are defined. The operations admit neutral elements, the ‘vacuum’ V and the ‘universe’ U, such that A V = A — A • U for all A in the system, and it is postulated that A A = A = A • A. The characteristic fea- ture of the algebra is the postulate of ‘absorption’: if A + B = B , then A ■ B = A, and conversely for all A, B in the system. Thus the algebra is essentially what G. Birkhoff (1934) called a lattice. Birkhoff also (1934) independently reduced projective geometry to a topic in lattice algebra, and Menger (193o) published a somewhat amplified account of his theory of 1928, in which it had been stated that the algebra is applicable to the theories of measure and probability. So far as the reduction of projective geometry to lattice algebra is concerned, this now seems an inevitable but unanticipated climax of the thorough- EMERGENCE OF STRUCTURAL ANALYSIS 263 going axiomatization of projective geometry — not very well known in Europe, apparently — by 0. Veblen and J. W. Young in 1910. The first two decades of the twentieth century witnessed an unprecedented activity in axiomatics, especially in the U.S.A., consequent on the work of D. Hilbert (1862-1943, German) in the foundations of geometry. The first volume of the Projective geometry (1910) at Veblen and Young subjected projective geometry to a logical rigor it had not experienced before, recon- structing this department of geometry as a hypothetico-dcduc- tive abstract system in accordance with Hilbert’s general formalistic program for all mathematics. Although geometric intuition may not have been abandoned in the rigorous treat- ment, it was not recognized, either officially or unofficially. The opening sentences of the work assert that Geometry deals with the properties of figures in space. Every' such figure is made up of various elements (points, lines, curves, planes, surfaces, etc.), and these elements bear certain relations to each other (a point lies on a line, a line passes through a point, two planes intersect, etc.). The propositions stating these properties are logically interdependent, and it is the object of geometry' to discover such propositions and to exhibit their logical interdependence. A more inclusive but somewhat vaguer description of geom- etry by one of the authors cited will be reported in a later chapter. The above passage is sufficient for the present; it almost begs to be translated into the language of lattices. In the expressive slang of 1945, it is a natural for such transla- tion by anyone who has ever glanced at the postulates for a lattice. Yet essentially these postulates were accessible ten years before the above manifesto of the aims of geometry was printed, and it was not until a quarter of a century after it appeared that the connection between lattices, or Dedckind’s dual groups, and geometry was perceived. But again this overlooking of what is now plain almost to the verge of truism is no reflection on the perspicacity of creative geometers. Rather is it merely another instance of the historical commonplace, emphasized long ago by W. Bolyai in connection with the final emergence of non-Euclidean geometry after centuries of seemingly unneces- sary struggle, that mathematical discoveries, like the spring- time violets in the woods, have their season which no human effort can retard or hasten. A further quotation from the Projective geometry of 1910 disposes, in the current mathematical manner, of certain debates 264 THE DEVELOPMENT Of MATHEMATICS, on the nature of ‘space’ that have exercised metaphysicans for many centuries: “Since any defined element or relation must be defined in terms of other elements and relations, it is necessary that one or more of the relations between them remain entirely undefined; otherwise a vicious circle is unavoidable.” Although, as will be seen in connection with ‘space’ of any finite number of dimensions, it is not necessary to choose ‘points’ as the ultimate undefined elements of geometry, they usually were chosen and frequently still are. Lines, etc., then became classes of points. Geometry, even of the elementary-school kind, was never primarily concerned with the elements in these classes, but with the intersections and unions of the classes. The lines, surfaces, etc., were the distinguished subdomains of the initially structureless chaos of points with which geometry actually dealt. However, points were usually in the background, if not in the mind, of the geometer doing geometry — implicitly Veblen’s definitions of ‘geometer’ and ‘geometry’ — ever since the time of Euclid. According to Euclid, “ a point is that which has no parts and has no magnitude.” But this nihilistic attempt at a definition was not incorporated into the deductive develop- ment of geometry until, as noted by Menger (1935), the lattice reformulation of both projective and affine geometry made logical use of Euclid’s definition in the deductive treatment of geometry. Euclid and his successors omitted to state what they meant by ‘part’; Menger supplied the deficiency by a precise logical definition. By corresponding explicit statement and inclusion of all hypotheses underlying his proofs, Menger showed that “great parts of projective and affine geometry may be developed upon the basis of ‘trivial’ axioms such as, for example, the postulates of [intersection and union as defined in Boolean algebra]. The outcome a mounted to the reduction of projective geo- metry to the alg'ebra of a suitably specialized lattice. The appropriate lattice 1 ? elements represent the several geometric objects or configurations of historical or mathematical interest, such as points, linehs, planes; the intersection of two configura- tions, in the Boolean sense, is the geometric intersection, or the configuration comntion to the two; the union of two configura- tions is the least 1 inclusive configuration containing both. Dcdekind’s axiom (ifor lattices, noted earlier), with a suitable finiteness condition t classifies the elements dimensionally by the corresponding Jordan-H older chains. In contrast to the EMERGENCE OF STRUCTURAL ANALYSIS 265 previous axiomatizations by Hilbert and by Yeblen and Young, the lattice representation distinguishes no configuration as basic; all enter the theory symmetrically. To include affine geometry, Mengcr (1935) imposed on lattices a reasonable axiom of parallelism. He was thereby enabled to develop projective and affine geometry together. In G. BirkhofFs reduction (1934), projective geometries were correlated with Boolean algebras, fields and rings of point- sets, systems of normal subgroups, systems of ideals, modules of a modular space, and systems of subalgebras of abstract algebras. There were also further correlations with the reduction of group representations, semi-simple hypercomplcx algebras, and compact Lie groups. All of these were to be expected from the reduction of projective geometries to lattices of certain types, and from the earlier instances of lattices given mostly by G. Birkhoff himself. From a philosophical point of view, perhaps the most interesting consequence of the lattice (or structure) represen- tation of projective and affine geometries was a possible re- percussion on the perennial speculations of metaphysicians concerning the nature of space. The debate of the lS20’s on the real spacial existence of the ideal elements of projective geom- etry has been mentioned, and it was suggested that the disputed topic had no meaning in the sense intended by the debaters. The intimate connections between the Boolean algebra of logic, projective and affine geometries, and lattices at least hint that the debaters may have been discussing merely the ingrained habits of their own reasoning processes, inherited for thousands of years as a legacy from the two-valued logic in which, it appears, human beings reason with a minimum ol thought. ‘Space’ may be nothing more mysterious than a triviality of a rudimentary logic. Retrospect and prospect Having rapidly traveled one main highway of modern mathe- matics as far as 1945, we may now glance back over our route and note certain landmarks of more than local — arithmetical or algebraic — interest. This has indeed been our objective from the start: to observe those changes in spirit and point of view which distinguish the whole of mathematics since 1900 from that of the nineteenth century. Vast empires of mathematics, where hundreds of assiduous 266 THE DEVELOPMENT OF MATHEMATICS workers have toiled and where many still labor, have not been even noticed. A detailed survey of the territory traversed would fill many volumes but, so far as may be judged at present, a complete account would neither add to nor subtract from two conclusions, evidence for only one of which has so far been presented. Both may be falsified tomorrow by the advent of another Descartes or a modern Gauss, or by a successor of Galois or Abel. The mathematics of the twentieth century differs chiefly from that of the nineteenth in two significant respects. The first is the deliberate pursuit of abstractness, in which relations, not things related, are the important elements. The second is an intense preoccupation with the foundations on which the whole intricate superstructure of modern mathematics rests. It maybe hazarded as a very problematical guess that when the history of mathematics is written a century hence, if mathematics is to last that long, the early twentieth century will be remembered chiefly as the first great age of healthy skepticism in mathe- matics as in much else. The nineteenth century in mathematics seems in retrospect to be of a piece with the rest of that smugly optimistic age. God was in his Heaven and all was right with the world. European civilization shipped its blessings wholesale to the heathen of all continents. There seemed to be no limit to the quantity of flimsy goods that could be manufactured and dumped at a handsome profit on the unenlightened who w r ere unable to dis- tinguish tin plate from sterling silver, or brass nose rings from solid gold. Nor v r as there any restraint in the output and con- sumption of mathematics. Nearly everybody appears to have believed that nearly everything was sound beyond all doubt. With the turn of the century a period of criticism and revalu- ation set in, and all but unteachable reactionaries agreed that the change was a decade or more overdue. The origins of both the abstract method and the critical approach can be traced definitely to the 1880’s. Neither attracted much attention till Hilbert in 1899 published his work on the foundations of geometry, and about the same time pointed out the basic importance for all mathematics of proving the self- consistency of common arithmetic. But it seems just to attribute the initial impulse to G. Peano (1858—1932, Italian) in his postu- lates for arithmetic (1889). Resuming the Euclidean program, Peano undertook the deduction of common arithmetic from an EMERGENCE OF STRUCTURAL ANALYSIS 2 67 explicitly stated set of postulates which were as free of con- cealed assumptions as he could make them. The postulational method is the source of both the modern critical movement and abstractness. In the least flattering light, both criticism and abstraction reflect the leaden hue of decadence. Viewed thus, the mathe- matics of the twentieth century is a modernized version of the Alexandrian age of criticism and sterile commentary that were the lingering death of Greek mathematics. Even if this should prove to be a correct diagnosis of twentieth-century mathe- matics, it docs not necessarily follow that mathematics is about to expire. Although they were long in coming, Archimedes had successors. Looking more sympathetically at the mathematics of the twentieth century, as nearly all professionals do, we see it full of life and more vigorous than ever. Criticism is necessary to see exactly what is sound so that the next step may be taken with reasonable safety. The abstract, postulational method is not mere cataloguing and pigeon-holing. It also is creation, but of a kind more basic than the disorderly luxuriance of the nineteenth century. Unless the enormous accumulations from that most prolific century in mathematical history arc sorted out and reduced to manageable proportions, mathematics will be smothered in its own riches. In the process of putting order into the huge mass by the abstract method, it is seen that much can be ignored. Should any of these neglected acquisitions ever be required, they arc now obtainable with much less labor than formerly. On the creative side, the postulational analysis of mathematical systems suggests innumerable new problems, some of which may be worth detailed investigation. Probably few engaged in this work of revaluation, simplifi- cation, and generalization imagine that an end has been reached in any direction. If Dedekind’s dual groups passed all but unnoticed for over thirty years, is it credible that all the promis- ing approaches have been explored, or that others will not be unexpectedly come upon ? A historical detail which was omitted in our rapid survey may be recalled here to suggest at least the possibility of progress in directions not yet followed. One of the clues that led Dcdekind to his creation of ideals was the theory of composition of binary quadratic forms, which is reflected in the multiplication of ideals in a quadratic field. Gauss systematized this theory of composition, and had cssen- EMERGENCE OF STRUCTURAL ANALYSIS 269 L. J. Mordell (1888-, U.S.A., England) proved (1942) that the preceding equation has no solutions with a-, y, z, quartic poly- nomials in one parameter with rational coefficients unless it = a 3 or n — 2a z , where a is a rational number; and B. Segre (Italian, England) made (1943) extensive applications of alge- braic geometry to polynomial diophantine equations. There is the possibility, however, that the problem of arith- metical forms will lose its interest for mathematicians. In other words, it will cease to be considered important. But importance sometimes, humanly enough, is only a tribute to the self- esteem of an egocentric mathematician. The problems which he can solve are, by definition, important; those which baffle him arc unimportant. To say that a particular problem has lost its importance for modern mathematics may therefore be merely a rationalized confession of incapacity. Until a method has been devised for solving a problem, or for proving that it is unsolvable if such be the case, common professional pride would seem to demand that it be considered. If this point of view is justified, the conclusion is that neither algebra nor arithmetic has reached an end in the modern ab- stract method, although that method may be a significant pre- lude to what our successors will create. We shall see that physics suggests the like in analysis. In any event, the beautiful achieve- ments of the method would have delighted Euclid, who first of all mathematicians produced a rounded example of the postula- tional technique. He may not have realized what he was doing, as a modern mathematician sees his work, but he did it. Pythag- oras, on the other hand, would have stood bewildered before the modern concept of number. lie would want to know what had become of the natural numbers 1, 2, 3, ... . All this time wc have been taking them for granted, not questioning their specious simplicity. Wc must now return to these so-called natural numbers — Kroneckcr’s gift from God — and sec what happened to them while number was disporting itself in a heaven Pythagoras never dreamed of. This will supply us with the link connecting algebra and arithmetic with analysis. Having examined this, we shall be in a position to proceed in later chapters to geometry and applied mathematics. Finally we shall be led back to the foundations of the whole structure, to see in them what our successors may designate as the characteristic distinguishing the mathematics of the twentieth century from all that pre- ceded it — critical, constructive doubt. CHAPTER 12 Cardinal and Ordinal to 1902 Lecturing in 1934—5 to his Chinese students in Peiping on the theory of functions of real variables, a distinguished Ameri- can analyst 1 observed that “The student has thus far taken the system of real numbers for granted, and worked with them. He may continue to do so to the end of his life without detriment to his mathematical thought. . . . On the other hand, most mathematicians are curious, at one time or other in their lives, to see how the system of real numbers can be evolved from the natural numbers.” The natural numbers are the positive ra- tional integers 1, 2, 3, .... A little earlier, a distinguished German analyst, 2 writing for his beginners in analysis, made an unusual request: “Please forget all that you have learned in school; because you have not learned it.” He was referring to such simple matters as 1 + 1 =2. Subscribing to all of these sentiments, we shall indicate the main steps by which mathematicians in the second half of the nineteenth century reached the modern concept of real numbers. The real numbers are the soil in which the classical theory of functions grows and flourishes and, as remarked above, are usually taken for granted by students. Also by some others. To anticipate, theories were constructed deriving the real numbers from the natural numbers. It was then sought to de- rive the natural numbers from something yet more basic, the theory of classes as in mathematical logic. By another of those curious coincidences regarding dates, what seemed like finality in this direction was reached in the closing years of the nine- teenth century. Those years, in more senses than one, were the end of a great epoch. 270 CARDINAL AND ORDINAL TO 1902 271 It is still no doubt true that students may take the real numbers for granted, as they did in 1902, without detriment to their mathematical thought. But it is no longer true that the more basic natural numbers can be taken for granted by any- body as they were by nearly everybody in 1902. Drowsy intui- tion has been shocked awake since the close of the nineteenth century; and the program of Eudoxus, resumed in the late nine- teenth century by the founders of the modern real number system, gave way in the twentieth century to another, more fundamental than any imagined by the Greek mathematicians. The center of interest shifted here as it is always shifting in mathematics. Indeed, it might have been mathematics and not an insignificant minor planet of a second-rate sun that Galileo had in mind when he muttered — according to a legend which should be true even if it may not be — “And still it moves,” as he rose from his knees and bowed to the Grand Inquisitor. Nobody yet has succeeded in stopping mathematical progress, in 1902 or in any other climactic year, as Joshua stopped the motion of a heavenly body at the battle of Gibeon. Equivalence and similarity The modern number concept is the link connecting the arithmetic and algebra of the past with the analysis, geometry, and mathematical logic of the present. Like the typical student of analysis today, we have been taking for granted the system of natural numbers from which, by successive generalizations, evolved the system of complex numbers, which in its turn sug- gested the hypercomplex numbers of modern algebra. Our im- mediate concern in this chapter is to indicate the main steps by which mathematicians in the latter half of the nineteenth cen- tury sought to ‘arithmetizc’ analysis. In the following chapter, we shall reach the same goal by a different route, that of the calculus from Newton and Leibniz to the year 1900; and we shall see again that the close of the nineteenth century marks a definite terminus in one direction of mathematical thought. This terminus, as will appear in the concluding chapter, v/as a turning point in the evolution of mathematics comparable in significance to that in the fourth century b.c., when Eudoxus parted company with the Pythagoreans. Once more the occasion for the new departure was the nature of irrational numbers. But in the modern pursuit of number, a far deeper spring of mathematical knowledge than any that refreshed the Greeks 272 THE DEVELOPMENT OF MATHEMATICS was tapped. The natural numbers had been taken for granted and it was found that they too presented subtle obstacles to clear understanding. The point of cardinal importance to be noted in the following summary account is the precise nature of the subtlest of these obstacles, which was perceived with dramatic suddenness only in 1902. The modern attack on number was directed against two closely related objectives: that of rigorizing the concepts of function, variable, limit, and continuity in analysis; that of penetrating the logical disguise of number. The first eventuated in the retreat from intuitive ideas of the calculus sublimated from unanalyzed conceptions of motion and continuous curves; the second culminated in the identification of cardinal numbers with classes. In both, the concepts of equivalence (or similarity) of classes, especially for infinite classes, played a dominant part. It is a matter of great historical interest (as indicated in an earlier chapter), that equivalence for classes was firmly grasped as early as 1638 by Galileo, 3 just a year after Descartes published his geometry. Galileo’s work was translated into English in 1665, the year that young Newton, rusticating at Woolsthorpe, thought out his first calculus. To us it seems strange that so plain an indication as Galileo’s of a feasible attack on all matters pertaining to the infinite was not pursued sooner than it was. But there is the earlier parallel of the Greek indifference to Babylonian algebra to suggest that mathematics does not always follow the straightest road to its future. As it would be difficult to find a clearer, more graphic state- ment than Galileo’s of the critical points, we quote 4 what he puts into the mouths of his characters, the sagacious Salviatus ( Salv .), and the questioning Simplicius (Simp.). The talk has been about the “Continuum of Indivisibles.” Salv. ... an Indivisible, added to another Indivisible, produceth not a thing divisible; for if that were so, it would follow, that even the Indivisibles were divisible. . . . Simp. Here already riseth a doubt, which I think un re solvable. . . • Now this assigning an Infinite bigger than an Infinite is, in my opinion, a conceit that can never by any means be apprehended. To make the infinite plain even to Simplicius, Salviatus patiently explains what a square integer is before proceeding as follows. CARDINAL AND ORDINAL TO 1902 273 Salv. Farther questioning, if I ask how many are the Numbers Square, you can answer me truly, that they be as many, as arc their propper roots; since every Square hath its Root, and every Root its Square, nor hath any Square more than one sole Root, or any Root more than one sole Square. This is the kernel of the matter: the one-one correspondence between a part of an infinite class, here that of all the natural numbers, and one of its subclasses, here that of all the integer squares. Continuing the argument, Salviatus compels Simplicius to surrender. Simp. What is to be resolved on this occasion? Salv. 1 sec no other decision that it may admit, but to say, that all Numbers arc infinite; Squares are infinite; and that neither is the multitude of Squares less than all Numbers, nor this greater than that: and in conclusion, that the Attributes of Equality, Majority, and Minority have no place in Infinities, but only in terminate quantities. . . . In modem terminology, two classes which can be placed in one-one correspondence are said to be equivalent 1 or similar. 5 In Galileo’s example, the class of all square integers is equivalent to the class of all positive integers. Again, a part (strictly, proper part) of a class C is any class which contains some but not all members of C, and nothing else. Galileo’s example shows that a class may be equivalent to a part of itself. A class is defined to be infinite if it is equivalent to a part of itself; and a class which is not infinite is defined to be finite. This means of distinguishing between finite and infinite classes was postulated 5 by B. Bolzano (1781-1848, of Prague), philosopher and theologian, and is basic in the modern theory of classes, both finite and infinite. Without it, Cantor’s theory of sets of points, fundamental in modern analysis, would not exist. It is interesting to note that Leibniz pointed out the simi- larity of the classes of all natural numbers and all even natural numbers, but drew the incorrect conclusion, rectified in Cantor’s theory, that “the number of all [natural] numbers implies a contradiction.” 7 Arithmetized analysis This is not the place to expound the theories of real numbers constructed by Cantor, Dcdckind, and Wcicrstrass; and we shall merely recall the few fundamental concepts necessary to give point to the historical climax of 1902. Wc observe first that the concept of a class (set, aggregate, assemblage, ensemble, Mcnge) was taken as intuitive in the preceding section. That ‘class’ is 274 THE DEVELOPMENT OF MATHEMATICS by no means an intuitive notion was recognized by Cantor, who in 1895 defined it thus: “By a class (Menge) we understand anv summary (Zusammenfassung) into a single whole of definite well-distinguished objects of our intuition (Anschauung) or of our thought (Denkens).” Possibly the exquisitely modulated philosophical German 5 is untranslatable into any blunter lan- guage, or incomprehensible to any but the initiated. With hesita- tion, then, we offer the following substitute in crude American ? “A class is said to be determined by any test or condition which every entity (in the universe considered) must either satisfy or not satisfy.” It seems clear to the unphilosophic mind that either of these definitions somewhat rashly invites philosophers to philosophize; and indeed, the invitation was accepted with alacrity. Whether this was a happy issue for the analysis of the nineteenth century out of all its afflictions seems to be in some doubt among professional analysts. 10 Another fundamental point in Cantor’s theory is the radical distinction between cardinal and ordinal numbers. For finite classes and numbers the distinction is almost trivial. Finite classes have the same cardinal number if and only if they are similar. Note that this does not define ‘cardinal numbed; it defines ‘same cardinal number,’ a significant distinction. It is quite possible to know that two criminals have the same name without knowing what the name is. The symbol 1, or 2, or 3, . . . denoting a cardinal number (not yet defined!) of a class is a mere mark or tag which is characteristic of the class without reference to the order in which its members are arranged. When the members of a finite class are counted in a given order, the mark or label 1 being assigned to the first, the mark 2 to the next, and so on, an ordinal number is correlated with each ele- ment of the ordered class; and if n is assigned to the last, n is also the mark denoting the cardinal number of the class. But ior infinite classes, as shown by Cantor, the like is no longer true; the marks for (transfinite) cardinals and ordinals differ, and the distinction between cardinal and ordinal is not trivial. The cardinal number of anv given class, finite or infinite, was defined by F. L. G. Frege (1848-1925, German) to be itself a class, namely, the class of all those classes similar to the given class. Thus the familiar cardinals 1, 2, 3, . . . of our unleame youth have vanished in the aO-nesses of infinities of classes con- taining respectively ‘one’ thing, ‘two’ things, and so on ior as many things as there may be in our Anschauung or in our CARDINAL AND ORDINAL TO 1902 275 Dcnkcns. This outcome may seem rather disappointing at first. But on prolonged reflection we are forced to agree with E. Landau (1877-1938, German) that what we learned at school we did not learn. Any class similar to the class of all the natural numbers is said to be denumerable or countable. Resolving Simplicius’ doubt about the conceit of “assigning an Infinite bigger than an Infinite,” Cantor proceeded to describe any desired number of such bigger Infinites. First, there is said to be no difficulty in imagining an ordered infinite class; the natural numbers 1, 2, 3, . . . themselves suffice. Beyond all these, in ordinal numeration, lies co; beyond cs lies w + 1; then u + 2, and so on, until co2 is reached, when w2 -j- 1, ufl -f 2, . . . are attained; beyond all these lies to 2 , and beyond this, or + 1, and so on, it is said, indefinitely and forever. If the first step — after which all the rest seems to follow of itself — offers any difficulty, we have but to grasp the scheme 1, 3, 5, , 2« + 1, . . . |2, in which, after all the odd natural numbers have been counted off, 2, which is not one of them, is imagined as the next in order. One purpose of Cantor in constructing these transfinitc ordinals, os, to + 1, . . . was to provide a means for the counting of well-ordered classes, a class being well-ordered if its members are ordered and each has a unique ‘successor.’ For cardinal numbers also Cantor described “an Infinite bigger than an Infinite” to confound the Simpliciuses of mathe- matics and enchant the Salviatuses. He proved (1874) that the class of all algebraic numbers is denumerable, and gave (1878) a rule for constructing an infinite non-denumcrabic class of real numbers. Were we to make a list of the spectacularly unexpected discoveries in mathematics, these two might head our list. Cantor’s proof is strictly one of existence. Providing no means for constructing any of the infinity of transcendental numbers whose existence is demonstrated, Cantor’s proof is in the medieval tradition of submathcmatical analysis. It would have convinced and delighted Aquinas. J. Liouville (1809-1S82, French), on the other hand, invented a method (1844) for constructing any one of an extensive class of transcendental numbers. His numbers were the first to be proved transcendental; Hcrmitc’s proof of the transcendence of e( — 2.718 . . .) followed in 1S73; F. Lindc- mann’s (1852-1939, German) forrr in 1882. Hinting at the con- troversies to come in the twentieth century, Kronecker demanded of Lindcrnann, “Of what value is your beautiful proof, since irrational numbers do not exist?'' We shall return to 276 THE DEVELOPMENT OF MATHEMATICS KroneckeP s program of arithmetization presently. It was quite different in both aim and scope from that of Cantor, Dedekind and Weierstrass in their project of arithmetizing analysis. In passing, we note that A. Gelfond in 1934 proved the transcen- dence of a b , where a is any algebraic number ^0, 1, and b is any irrational algebraic number. In the program of arithmetizing analysis, the rational numbers presented no difficulty. By the device of number- couples subjected to appropriate postulates, the properties of positive rationals were referred to those of positive integers, and negative rationals were driven back with equal ease to positive rationals. Thus all the rationals were derived by a simple routine from the natural numbers. Proceeding to infinitely the greater part of the continuum of real numbers, Cantor defined irrationals by infinite sequences of rationals; for example, may be defined by the sequence 1, 14/10, 141/100, 1,414/1,000, 14,142/10.000, .... Generally, if a i} a~, a z , . . . is any infinite sequence of rational numbers which is such that, for each rational e > 0, however small, there is an index m such that \a n — a\ < e for every n , v Se m, the sequence is said to be regular. It is postulated that every regular sequence defines a number; the class of all so-defined numbers is the real number system. With suitable definitions of equality', greater, less, sum, difference, product, and quotient, it was shown that these num- bers satisfy the requirements of experience. In particular, a meaning was given to the formalism of such useful statements as V2 X A/3 = V2 X 3, V2 X VI = V3 X VI. Cantor had arith- metized the continuum of real numbers. Geometry also shared in the benefits which arithmetization had conferred on analysis. A one-one correspondence between all the points on any segment of a straight line and the con- tinuum of real numbers was established. This done, C. Jordan (1838-1922, French) banished intuition from the conception of curved lines by giving a strictly arithmetical definition 11 oi a curve as a plane set of points which can be put in one-one correspondence with the points of a closed segment [a, b). This seems like restating a platitude in pedantic obscurity when we cite its simplest example, the parametric equations x = r cos 1, y = r sin t of the circle x" -r y~ = t~. It seems rather less plati- tudinous when we recall that Peano (1890) constructed a re^ continuous plane curve, as the locus of a point {x, y) >’hose ~ _ if A = g( t ) y-ith /, g uniform- coordinates are given by x CARDINAL AND ORDINAL TO 1902 277 continuous functions of the real variable t in the range 0 g / g 1, completely filling the square 0 g x ^ 1, 0 y g 1. In fact, he described two such curves passing through every point of the unit square. Many more examples of such ‘space-filling curves’ have been constructed since Peano exhibited the first; and what in 1890 appeared as a collapse of the geometrical heavens has reappeared as a commonplace phenomenon in Ph.D. disser- tations. “And still it moves.” Equally unexpected miracles began illuminating the con- tinuum itself. By a fairly immediate gencraliaation, classes (sets) of points in a continuum (‘space’) of any finite or of a denumer- ably infinite number of dimensions were invented. Cantor proved that in each instance all the points in the whole space can be put in one-one correspondence with all the points on any straight-line segment. In a plane, for example, there are precisely as many points on a segment an inch long as there are in the entire plane. This, of course, is contrary to common sense; but common sense exists chiefly in order that reason may have its Simpliciuses to contradict and enlighten. However, some Simplicius occasionally interjects a shrewd objection, upsetting the even progress of the discussion; and if he but seldom gets the better of an argument, he can at least cause it to stumble badly. Kronecker elected himself the Simplicius of Cantor, Dcdekind, and Weierstrass. His objections will be noted presently. A profound question which exercised Cantor’s utmost powers was this: Can the continuum of real numbers be well ordered? In 1883 he thought he had answered this affirmatively. Objections to his attempted proof were largely responsible for the descent of mathematics after 1900 to deeper levels in its efforts to escape deceptive intuition. Another problem which baffled Cantor was to prove or disprove that there exists a class whose cardinal number exceeds that of the class of natural numbers and is exceeded by that of the class of real numbers. ’1 his problem seemed to be still open ,: in 1945. Whatever may be the ultimate fate of Cantor’s theories of the infinite, continuity, and the number system, it appears likely that he will be remembered with Eudoxus as one of those who breached what is, after all, the central fortress of mathematical analysis. So also will Dcdekind and Weierstrass. Like Cantor, these two also derived the number system of analysis from the natural numbers. Dcdekind by his device of cuts, Weierstrass by classes of rationals. Another analyst who reached the same goal 2 78 THE DEVELOPMENT OF MATHEMATICS was C. Mera y (1835-1911, French); but possibly the difficulty of his exposition deprived him of his just share of fame. One or other of these theories is so familiar today to every student of an advanced course in the calculus that there is no need to describe them here. They are open to precisely the same objections that mathematical logicians have raised to Cantor’s Mengenlehre. But in stating this plain matter of fact, we do not imply that these theories have been rejected as totally wrong or barren. They still afford the most promising approach to an under- standing of numbers and their part in analysis. If a theory is imperfect, that is perhaps only because it is not yet either dead or useless. In the following chapter we shall see that the analysis of trigonometric (Fourier) series was partly responsible for the attempt to put a firm logical foundation under the continuum of real numbers. Many contributed significantly to the attempt; but the four whose work has been noted were the earliest to see clearly what needed doing and the first to attempt it. Of the many who prepared for the final success, one may be mentioned here, P. du Bois-Reymond (1831-1889, German), partly for his own subtle researches in analysis, partly because it was owing to his insistence that Weierstrass permitted a most disconcerting invention of his own to become public. Intuitively, a continuous arc of a curve has a tangent at every point on the arc; Weierstrass constructed the equation of a continuous curve having no tangent at any of its points. He is said to have com- municated this to his circle in 1861, but for some reason withheld it until du Bois-Reymond in 1874 asked him whether such a curve was possible. This example alone showed the necessity for some rigorous theory of the real number system to replace the pernicious intuitions which had seeped into analysis from geometry and kinematics. Existence and constructibility Kronecker has already been mentioned for his technical con- tributions to algebra and the higher arithmetic. These probably represent his choicest creations to those who have taken the pains to appreciate them; but Kronecker is more widely known to the mathematical public for his philosophy of mathematics. At one time he was looked upon by some of the analysts, including Weierstrass, as a sort of personal devil. It was feared that Kronecker’s philosophy was wholly destructive; and it cannot CARDINAL AND ORDINAL TO 1902 279 be gainsaid that he hated the highly speculative analysis of his famous contemporaries. If Kronecker spelled Satan to Cantor, Cantor signified the personification of all mathematical evil to Kronecker. Dcdckind’s definition of irrationals as cuts in infinite classes of rationals, Cantor’s sequences of rationals defining irrationals, and Weierstrass’ irrationals as classes of rationals, all ultimately referred the continuum of real numbers to the natural numbers. The ‘magnitudes’ of Eudoxus were replaced by hypothetical constructions performed upon the numbers 1, 2, 3, ... . Thus the arithmetization of analysis was a return to the program of Pythagoras. Mathematical mechanics having been reduced to a department of analysis, it too was potentially arithmetized, at least by implication, and likewise for geometry. All at last was reduced to number as the Pythagoreans imagined number, but at a cost which they would never have attempted to meet, infinities upon infinities. As thoroughgoing a Pythagorean as Pythagoras himself, Kronecker insisted that the infinities be banished, and that all mathematics be built up by finite constructions from the natural numbers. Unless a mathematical object were construc- tiblc in a finite number of non-tentative steps, it did not exist for Kronecker, no matter how many and how rigidly logical the transcendental proofs of its existence. 15 Such a philosophy of mathematics did not make nonsense of analysis as Kroncckcr’s rivals had re-created it; analysis was simply abolished. As if to lend some plausibility to the destructive part of Kroncckcr’s program, flagrant contradictions began appearing in the late 1890’s in reasoning of apparently the same general character as that used by the arithmetizers of analysis. Twenty- four centuries after Zeno’s runner had lost his race, his heirs appeared on another course, fresher and fleeter of foot than that ancient had ever been. The new antinomies of the infinite sprang from the protean ‘all’ by which the irrationals were generated in the arithmetization of analysis: ‘all’ natural numbers; the class of ‘all’ rational numbers whose squares arc less than 2, and the class of ‘all’ those whose squares arc greater than 2 as the ‘cut’ defining VI, and an infinity more. The first of the new and more vigorous paradoxes was fathered in 1897 by the Italian mathematician C. Burali-Forti: the well-ordered scries of all ordinal numbers defines a nau ordinal number which is not one of the all. 280 THE DEVELOPMENT OF MATHEMATICS A less technical parados on ‘all’ was B. A. W. Russell’s (1872-, English) of 1902: Is the class of all those classes which are not members of themselves a member of itself? Either Yes or No leads to a contradiction. The same irrepressible successor of Zeno recorded a yet simpler ‘all’ paradox: A barber in a certain village shaves all those, and only those, who do not shave themselves; does the barber shave himself? There are many- more. All — if we may use the word without danger of engender- ing another exasperating paradox — or, if not all, then many conceal a dubious ‘all.’ One root of the technical mathematical difficulties has been traced to the concept of ‘class’ itself, Cantor’s ‘Menge’ as defined by him. The attempt to be logically precise ended in hopeless confusion. The problems of the nineteenth century met those of the twen- tieth in the extraordinarily subtle mind of F. L. G. Frege (1848— 1925, German), part of whose lifework 14 was an endeavor to put a self-consistent foundation under the number concept. In 1884, Frege was led to his famous definition of the cardinal number of a given class as the class of all those classes similar to the given class. From this definition, Frege derived the usual properties of numbers familiar in common arithmetic. Unfortunately, to develop the subtlety of his reasoning with precision, he had found it necessary to clothe his proofs in a complicated dia- grammatic symbolism which repelled all but the hardiest and most obstinate readers. As a result, the epoch-making definition embedded in his work passed unnoticed by the mathematical public till Russell, by different reasoning, independently reached (1902 ?) the same definition and expounded it in English. Frege had used the theory of classes. The second volume of his masterpiece 14 appeared in 1903. It closes with the following confession. “A scientist can hardly encounter anything more undesirable than to have the foundation collapse just as the work is completed. A letter from Mr. Bertrand Russell put me in this position as the work was all but through the press.” Russell’s letter contained his paradox on the class of all classes that are not members of themselves. Frege’s pessimism is understandable enough. But it was to prove unjustifiable in the long view of mathematical progress. The attempt to found the number system on a theory of classes seemed to have failed, and no doubt it had, at least temporarily. By the collapse of the class theory of the number system, analysis was left without a foundation, and hung suspended in mid-air CARDINAL AND ORDINAL TO 1902 281 like Mahomet’s coffin, sustained only by a miracle of faith. But the very failure revealed the nature of the fundamental weak- ness. A younger, more vigorous generation attacked the problem of bringing analysis down to reason again. Profiting by the experience of the nineteenth-century arithmetizers, the mathe- matical logicians of the twentieth century set themselves the task of putting a self-consistent foundation under all mathe- matics, not merely under analysis. Their efforts quickly carried the program of Leibniz for a strict symbolic reasoning far beyond anything he ever conceived, and in so doing created much new mathematics. In the meantime, analysts, geometers, arithmeticians, and algebraists continued their technical labors as if there were no ‘crisis’ in the foundations, creating interesting and useful things as their forerunners had done for centuries. Their confidence in the security of the essentials of their creations is justified by experience. Changing philosophies of mathematics may trans- form proofs and even theorems out of easy recognition as mathe- matics develops, and much is thrown away. But if history is a reliable prophet, 15 there will remain of the analysis of the nine- teenth century as much, relatively, as remains of Euclid’s proposition I, 47. CHAPTER 13 From Intuition to Absolute Rigor 1700-1900 In following the development of the number concept to its final phase in modern arithmetic and abstract algebra, we have caught occasional glimpses of the spirit of mathematics as it has become since the dose of the eighteenth century. Similar pro- found changes appear when we observe analysis. We now return to the eighteenth century and note the first attempts to construct a logically sound differential and integral calculus. The contrast between what passed for valid reasoning then and what is now demanded is violent. Passing back to the eight- eenth century, we find ourselves in a dead world, almost in another universe. Some of Newton’s successors who strove to make sense out of the calculus are among the greatest mathe- maticians of all time. Yet, as we follow their reasoning, we can only wonder whether our own will seem as puerile to our succes- sors a century and a half hence. It is not a question here of the enduring things these famous men did with their analysis in applied mathematics, or even of the basic algorithms which they invented and which have also lasted. We are concerned solely with their avowed attempts to put consistent meaning into the analysis itself. Two decisive turning points We saw that Newton himself was dissatisfied with his account of the fundamental concepts of the calculus. The like hoi s or 282 FROM INTUITION TO ABSOLUTE RIGOR 283 Leibniz, %vho half promised Huygens that some day he would re- turn to the beginning and set everything right. But he never did. After the passing of Newton and Leibniz, critics of both ap- proaches made themselves heard, and conscientious analysts, responding to legitimate objections, attempted to put a firm foundation under the calculus. Their efforts gradually disclosed the depth of the difficulties, and in the nineteenth century were partly responsible for the creation of vast new departments of mathematics, such as the theories of Dedekind and Cantor. We shall indicate the main stages in this extremely complex evolu- tion by which the calculus of 1700 developed into that of 1900. As wide a gulf separated the great analysts of the late eight- eenth century from those of the early nineteenth as severed the Pythagoreans from Eudoxus. After 1929 — the historic year in which the great depression began in the United States — another deep fissure opened up, cutting off retreat to the nineteenth century, apparently forever, when K. Godel re-examined the possibility of a consistency-proof for rational arithmetic. In following any account of the evolution of rigor in the cal- culus, it must be remembered that opinions on many unsettled points differ, sometimes widely. Further, it has been difficult for some to avoid reading their own more exact knowledge into work of their predecessors which, if taken at what seems its intended value, gives no hint that its authors were ever conscious of what later stood out as fatal defects. For example, the gener- ous J. lc R. d’Alembert (1717-1783, French) in 1770 ascribed to Newton a fully developed theory of limits that but few analysts today can detect in what Newton published. And last, before proceeding to details, we emphasize once more that in exhibiting the shortcomings in the work of the older analysts, there is no implication that perfection has been attained in our own. The mistakes and unresolved difficulties of the past in mathematics have always been the opportunities of its future; and should analysis ever appear to be without flaw or blemish, its perfection might only be that of death. Five stages The general trend from 1700 to 1900 was toward a stricter arithmetization of three basic concepts of the calculus: number, function, limit. More subtle questions concerning the meaning of ‘variable’ scarcely entered before the twentieth century'. In the period under discussion there were five well-marked stages, 284 THE DEVELOPMENT OF MATHEMATICS which may be easily retained by the names and dates of certain leaders in each. With the first are associated Thomas Simpson (1710-1761, English) in England, and G. F. A. l’Hospital (1661—1704, French) on the Continent. Euler (1707-1783, Swiss) represents the second stage; Lagrange (1736-1813) the third- Gauss (1777— 1855) and Cauchy (1789—1857) the fourth; and Weierstrass (1815—1897, German) the fifth. Euler is the culmina- tion of the almost wholly uncritical schools of Newton and Leibniz; Lagrange marks the earliest recognition by a mathe- matician of the first rank that the calculus was in a thoroughly unsatisfactory condition; Gauss is the modem originator of rigorous mathematics; Cauchy is the first modern rigorist to gather any considerable following; and Weierstrass, dying in 1897, epitomizes the progress made in exactly one hundred years from the first publication (1797) of Lagrange attempting to rigorize the calculus. The golden age of * nothing ’ ‘Formalism 5 in analysis means manipulation of formulas in- volving infinite processes without sufficient attention to con- vergence and mathematical existence. Thus the formal binomial theorem applied to (1 — 2 ) _1 gives -1 = 1+ 2-1-4 + 8 + 16+..., a meaningless result which did not astonish Euler, the great- est but not the last of the formalists. ‘Intuition 5 in analy- sis, as we shall use it here, means an unreasoning faith in the universal validity of what the senses report to the intellect concerning motion and geometrical diagrams. Newton was the greatest of the intuitionists in analysis, with the far more philosophical Leibniz a distant and therefore honorable second. (Both formalism and intuitionism in the mathematics of the twentieth century have different meanings, to be noted in the concluding chapter.) The direction of evolution in the calculus has been constantly away from formalism and intuition, al- though neither is yet extinct. The first and crudest stage is represented in England by the two editions (1737-1776) of Simpson’s classic Treatise on fluxions , in which 1 intuition flourishes freely and rankly. At- tempting to clarify Newton’s intuitive approach to fluxions through the generation of “magnitudes” by “continued mo- tion,” Simpson succeeded only in adding a deeper obscurity o 286 THE DEVELOPMENT OF MATHEMATICS behind a goodly part of these savage assaults on the work of eminent mathematicians. The shrewdest attack of all was delivered by a man who was not a mathematician and who made no claim to be, G. Berkeley (1685-1753, Irish): half-heir of Jonathan Swift’s Vanessa; at one time self-appointed apostle of culture to Bermuda but shipped by mistake to Newport, Rhode Island, where for three stagnant years (1728-31) he rusticated; later Bishop of Cloyne in his native Ireland; famed for his sub- jective idealism that out-idealized Plato, and immortal for his advocacy of tar water as a remedy for spiritual disorders and smallpox. A mind as keen as Berkeley’s was needed to expose once for all the subtle fallacies in Newtonian fluxions, and the sagacious bishop spared no logic in his withering attack. A philosophical amateur did what professional mathematicians had shown themselves either too partisan or too tender-minded to do. Although few professionals would admit that either corpse was dead, Berkeley slew both fluxions and ‘prime and ultimate ratios.’ Berkeley’s assault in his Analyst (1734) was not just another of the vulgar wrangles, like the controversy over priority in the calculus, that disfigure the career of the Queen of the Sciences. It was one of the ablest critiques the leading mathematicians of any period have ignored, possibly because it came from one who was not a member of their somewhat exclusive guild. For once a philosopher turned the tables on mathematicians by convicting the fluxionists of changing their hypothesis in the middle of an argument. Until Berkeley’s time, it had been supposed that this effective tactic in logomachy was an exclusive prerogative of dialecticians. Berkeley contended that substituting x + o for x in x n and letting o vanish in the final step, to get the fluxion of x n , is a shift in the hypothesis: “ . . . for when it is said let the increments be nothing, 4 or let there be no increments, the former supposition that the increments were something, or that there were increments, is destroyed, and yet a consequence of that supposition, i.e., an expression got by virtue thereof, is retained. ’ There were replies to this; but that which is unanswerable can- not be answered, and the controversy blew over, leaving scarcely a ripple on the muddy waters of mathematical analysis as they were in 1734. Berkeley’s criticisms were well grounded, but neither he nor they were taken seriously by the leading analysts of his time, and mathematics sought salvation in its own way. It is amusing FROM INTUITION TO ABSOLUTE RIGOR 287 to recall in passing that another question of salvation inspired Berkeley to his attack on fluxions. The full title of his work is The analyst: or, a discourse addressed to an infidel mathematician. Wherein it is examined whether the object, principles, and infer- ences of the modern analysis are more distinctly conceived, or more evidently deduced, than religious mysteries a7id points of faith. Only an Irish bishop who was also an idealistic philosopher could have conceived such a heroic project. It seems that Newton’s friend Halley, posing as a great mathematician, had proved conclusively to some deluded wretch the inconceivability of the dogmas of Christian theology. The converted one, a friend of Berkeley’s, refused the latter’s spiritual offices on his deathbed. This was in the very year that Berkeley became a bishop. Profoundly shocked by the soul-destroying savagery of “the modern analysis,” and mindful of his education in semicivilizcd Rhode Island, the good bishop went after the scalp of fluxions. He secured it; and the wretch who had been converted to infidelity by a nonsensical argument was avenged, although it may have been too late to save his soul. The triumph of formalism Euler’s almost total capitulation to the seductions of formal- ism is one of the unexplained mysteries in mathematics. Like Newton, Euler was aware that scries must ‘in general’ 5 converge if they arc to be practically useful, as in astronomy; but unlike Newton, he was unable to restrain himself this side of absurdity. Euler appears to have believed that formulas can do no evil; and so long as they continued to furnish their parent with ever new and more prolific variations of themselves, he encouraged them to increase and multiply, trusting no doubt that someday all their offspring would somehow be legitimized. Many of them have, and flourish today as lusty theories whose first bold steps were taken in several editions of three masterpieces of this most prolific mathematician in history: Jntroduclio in analysin infini- torum (174S); lnstitutior.es calculi diffcrcntialis (1755); Institu- tions s calculi integralis (176S-1794). The aim of the Introductio is to obtain by elementary means the utmost of what is so obtainable, but which is usually derived by the differential and integral calculus. The work is in two parts, an analytic and a geometric. Among the host of results are the expansions of the circular (singly periodic) functions, trans- formations of infinite products into infinite scries, and develop- 288 THE DEVELOPMENT OF MATHEMATICS ments into series of partial fractions. The last suggested one approach in the nineteenth century to elliptic (doubly periodic) functions. One chapter derives the basic formulas in the analytic- algebraic theory of partition of numbers. There are two heroes in this great drama of formalism, the expansion of f from the limit (Euler’s style) of (1 + x/n) n as n tends through positive values to infinity, and the cardinal formula of analytic trigo- nometry, e' x = cos x i sin x, i = ( — 1)*. Creative formalism such as this is responsible for the impatient criticism of ex- treme mathematical rigor as rigor mortis. The geometric part handles analytic geometry, both plane and solid, with equal freedom and complete mastery. The material includes special curves and surfaces, tangents and tan- gent planes, normals, areas, and volumes. Turning partly away from intuitionism, Euler abandoned geometry in the Institutiones. This work is remarkable for its exhibition of analogies between the infinitesimal calculus and the calculus of finite differences, and the use of the latter to approximate to results in the former. There is no hint of con- vergence, but slowly convergent series are converted in masterly fashion into others more rapidly convergent. Here also the usual formal parts of the differential and integral calculus are devel- oped in minute detail. One prophetic triumph of manipulative skill may be specially cited: Euler obtains the addition theorem for elliptic integrals as an exercise in differential equations. A function to Euler became a congeries of formal representa- tions transformable into one another by ingenious devices rang- ing from elementary algebra to the calculus. Glorying in the pragmatic power of his methods, Euler needed to see nothing absurd in his conception of the differential calculus as a process of determining the ratio of vanished increments. His differentials are first and last absolute zeros whose ratios by some incompre- hensible spiritualism materialize in finite, determinate numbers. As the usually courteous Lagrange observed, Euler’s calculus does not make sense. If the end ever justifies the means in analysis, Euler was justified. He sought beautiful formulas, and he found them in overwhelming abundance. But obviously the calculus could not continue indefinitely on the primrose path so happily followed by this boldest and most successful formalist in history. Even Euler caught an occasional whiff of the everlasting bonfire in the absurdities that floated up now and then from the pit just FROM INTUITION TO ABSOLUTE RIGOR 289 ahead of him. Others scented damnation more keenly than he, among them his friend d’Alembert. Best known for his principle in mechanics (1743), d’Alem- bert should be remembered also for having been the first (1754) to state 6 that “the theory of limits is the true metaphysics of the differential calculus.” That nobody in the eighteenth century carried out the implied program, or was capable of doing so, is beside the point; d’Alembert saw clearly that what the calculus needed was not more formulas but a foundation. He regarded Newton’s calculus of prime and ultimate ratios as a method of limits. Newton might have agreed with this had it been pointed out to him. Lagrange's remedy A new direction was taken by Lagrange in his ambitious Theoric dcs fonctions analytiques (1797, 1813), and his Calcul dcs f evictions (1799, 1806). These -were conscious attempts to escape from Euler’s conception of a function as a mere formula or al- gorithm, although Lagrange himself substituted another kind of formula, the power series, for the representation of all func- tions. His escape took him from one kind of formalism to another. Dissatisfied 7 with the efforts of all his predecessors and contemporaries, he rejected both infinitesimals and limits as being unsound, too difficult for neophytes, and, in the least complimentary sense, metaphysical. Lagrange was the leading mathematician of the eighteenth century and one of the greatest in history. Pie also was the first to restate Taylor's theorem with a remainder term. Keeping all this in mind we shall, if we have the least grain of caution in us, be extremely conservative in our estimates of current rigor when we remember what finally satisfied Lagrange. He based his calculus on the expansion of a function in a Taylor scries, assuming that, "by the theory of series,” }{x + h) — /(*) -p ah -f- bh 2 -p ch s + • • • • From this he convinced himself that if a s= "the derived function of /(.v),” then 2b — f'{x), where f"(x) — (f'(x))\ and so on, all, as he imagined, without benefit of limits. He points out 3 that anyone familiar with the usual form of the calculus will sec that/'(.v) is really df(x)/dx. But by what he has just deduced from "the theory of scries,” it is clear that df(x)/dx docs not in any way depend upon limits, prime and ultimate ratios, or in- 290 THE DEVELOPMENT OF MATHEMATICS finitesimals as formerly ;/'(#) is merely the coefficient of h in the expansion of f(x + h ) in ascending powers of h. Need more be said ? 9 Gains to 1800 The net gains in the eighteenth century appear to have been four. Berkeley disposed of fluxions and of prime and ultimate ratios. Euler produced a vast wealth of results by purely formal uses of the calculus; and so sure was his instinct for what was to remain valid that his work was the point of departure from which many of his more productive successors made some of their most significant advances. To cite only two instances, Gauss, Abel, Jacobi, and Hermite were indebted directly to Euler in their more rigorous work on the theta and elliptic functions; the Eulerian integrals suggested to Legendre, Gauss, and Weierstrass extensive developments in the theory of the gamma function. The third outstanding gain was d’Alembert’s demand that' the calculus be founded on the method of limits. The execution of this program had to wait for Cauchy (1821). The fourth gain was the hint implicit in Lagrange’s abortive attempt to generate the calculus from power series. Weierstrass, in his theory of analytic functions, carried out what might have been an eight- eenth-century program had Lagrange seen just a little more clearly what he was actually doing. Ridiculous interlude Formalism of a narrower sort than Euler’s reached its absurd climax in the period between Lagrange and Cauchy. Com- binatorial analysis, in the trivial sense of manipulating binomial and multinomial coefficients, and formally expanding powers of infinite series by applications ad libitum ad nauseamque of the multinomial theorem, represented the best that academic mathematics could do in the Germany of the late eighteenth century. The combinatorial school headed by C. F. Hindenburg (1741-1808, German) was the unlovely offspring of two human failings, neither of which is popularly supposed to have any relevance for the sublimities of pure mathematics: blind hero worship and national jealousy. The German Leibniz in his combinatorial analysis of the dis- crete had created a rival for the English Newton’s infinitesimal analysis of the continuous. Therefore, abandoning the calculus FROM INTUITION TO ABSOLUTE RIGOR 291 and its astronomical applications to the British, the Swiss, and the French, who had the lead already, the German mathemati- cians would loyally follow their national hero. Completely miss- ing the deeper significance of Leibniz’ program as a step toward the ‘universal characteristic,’ his patriotic disciples elaborated its superficialities in a wealth of useless formulas. The title of the masterpiece in this ambitious futility brazenly proclaimed the multinomial theorem to be the most important truth in the whole of analysis. 13 Still more grandiose pretensions to omnipotence were urged by an egocentric Pole, H. Wronski (177S-1S53), ardent envicr of Lagrange, also a disciple of the combinatorial school, although his transcendent conceit 11 denied any progenitor but Wronski for himself and his “Supreme Law” which, he insisted, con- tained all analysis, past, present, and future. Both Wronski’s claims and those of the combinatorialists have been disallowed by the supreme court of mathematical progress, from which there is no appeal. His criticism of Lagrange’s attempt at rigor was justified; but his own substitute was no better. The labors of this almost forgotten combinatorial school, however, were not without lasting benefit for the calculus. They filled young Gauss with such an intense disgust for formalism and all its works that he resolved to go his own lonely way and put some meaning into analysis, even if it cost him the patronage of every academician in Germany. He even favored the mighty Hindcnburg with an extremely sarcastic letter. In 1S12 Gauss published his classic memoir 1 - on the hyper- geometric scries, in which, for the first time in the history of mathematics, the convergence of an infinite series was ade- quately 1 ' 1 investigated. Others before Gauss had gone as far as stating tests for convergence, notably Leibniz, for alternating series, and E. Waring (1734-179S, English), who had given what is usually called Cauchy’s ratio test as early as 1776; but Gauss was the first to carry through a rigorous treatment. Intuition transformed From the calculus of Newton and Leibniz to that of Lagrange there is no indication that analysts were aware of the necessity for an understanding of the real number system. Nor is there in the next stage, that of Cauchy. Even as late as 1945 ‘quantities’ occurred frequently in the writings of professional analysts with no explanation of what a ‘quantity’ may signify. 292 THE DEVELOPMENT OF MATHEMATICS Imagining, perhaps, that he was banishing deceptive intui- tion forever from analysis, Cauchy succeeded in driving it down to a far deeper level where it might continue its subtle mischief unobserved. The crude visual and geometrical intuition of the early analysts was transformed into an uncritical faith in the logical possibility of the continuum of real numbers. Cauchy, Abel, and possibly Gauss 16 — for he seems to have left no record of his beliefs on this matter — adhered to this faith. The definitions of limit and continuity current today in thoughtfully written texts on the elementary calculus are sub- stantially those expounded and applied by Cauchy in his lectures and in his Cours d’ analyse (1821), his Resume des legons donnees a Vecole -polytechnique (1823), and his Applications du calcul infinitesimal a la geometrie (1826). The differential quotient, or derivative, is defined as the limit of a difference quotient, the definite integral as the limit of a sum, and differentials as arbi- trary real numbers. The continuity of a function and the con- vergence of an infinite series are referred to the concept of a limit. Thus Cauchy in effect created the elements of the classical theory of functions of a real variable. It was Cauchy’s rigor that inspired Abel on his visit to Paris in 1826 to make the banish- ment of formalism from analysis a major effort of his projected lifework. But, indicative of the subtleties inherent in consistent think- ing about the infinite and the continuum, even so cautious a mind as Cauchy’s went astray when it surrendered itself to intuition. He believed for a time that the sum of any convergent series of continuous functions is continuous, and that the integral of the sum is always obtainable by termwise integration. Later (1853, 1857) he recognized uniform convergence, discovered independently by the mathematical physicist G. G. Stokes (1819-1903, Irish) in 1847 and P. L. v. Seidel (1821-1896, German) in 1848. Cauchy also fell foul of the traps guarding interchange of limits in double-limit processes, as also did Gauss, 10 another plain hint that the real number system is less innocuous than it appears to naive intuition. A suggestion from physics It is rather surprising to find a main source of modern ngor in the work of a mathematical physicist who had almost a con- tempt 17 for mathematics except as a drudge of the sciences. J. B. J. Fourier (1758-1830, French) published his masterpiece, FROM INTUITION TO ABSOLUTE RIGOR 293 La thcoric analytiquc de la chaleur, in 1822, the year after Cauchy had rigorizcd the calculus. But if it had appeared twenty years after Cauchy’s lectures, it probably would not have differed materially from what it was. Fourier had obstinately refused for fifteen years to heed the objections of Lagrange and others that vital parts of his analysis were unsound. In his famous classic on the conduction of heat, 17 Fourier proved himself the Euler of mathematical physics. Leaving convergence to take care of itself, he trusted his physical intuition to lead him to correct results, as it usually did. The sixth section of Fourier’s Thcoric 17 is the one which con- cerns us here. It is devoted to the solution “of a more general problem, which consists in developing any function whatever in an infinite series of sines or cosines of multiple arcs. . . . We proceed to explain the solution.” 13 Having done so for a special case, Fourier continues, 15 “We can extend the same results to any functions, even to those which are discontinuous and entirely arbitrary'. To establish clearly the truth of this proposition, we must examine the foregoing equation,” which he does, in the manner of Eulcrian formalism. 50 The outcome is the expansion of an ‘arbitrary’ odd function in a sine series. Lagrange in 1766 had constructed by a process of interpolation a finite summation formula from which Fourier’s result is obtainable by a leap into infinity, but he “abstained from the transition from this sum- mation formula to the integration formula given by Fourier.” 21 Lagrange’s difficulty was that he had a mathematical conscience. Physical intuition supplied Fourier’s lack of mathematical inhibitions and guided him to the general statement of his famous theorem. The mathematical physicist’s boldness taught pure mathe- maticians several things of the first importance for the future of analysis. Purists gradually came to realize that their intuitions of ‘arbitrary’ function, real number, and continuity needed clarification. P. G. L. Dirichlct’s (1805-1S57, German) defini- tion 22 (1837) of a (numerical-valued) function of a (real, numer- ical-valued) variable as a table, or correspondence, or correlation, between two sets of numbers hinted at a theory' of equivalence of point sets. When G. F. B. Ricmann (1826-1866, German) in 1854 investigated 23 tire representation of a function by' a trigonometric (Fourier) scries, he discovered that Cauchy had been too restrictive in his definition of an integral, and showed that definite integrals as limits of sums exist even when the 294 THE DEVELOPMENT OF MATHEMATICS integrand is discontinuous. Later (date uncertain) he invented a function, defined by a trigonometric series, which is continuous for irrational values of the variable and discontinuous for rational values. 24 It was clear that the continuum of real numbers had not been thoroughly understood. With our present knowledge we see once more what Cantor was the first to perceive, the necessity for a theory of sets of points. Cantor’s investigations, like Riemann’s, began in Fourier series. The demand for clearer understanding of limits, continuity, and derivatives was further emphasized in 1874 by the publicity given to Weierstrass’ example of a continuous function having no derivative or, what is equivalent, of a continuous curve admit- ting no tangent at any point. Intuition all but expired. Such appear to have been the principal impulses behind the creation of the modern continuum. The unforeseen phenomena cited, and many others almost equally unexpected but of the same general character, seemed to indicate that all the diffi- culties were ultimately rooted in the real number system. Urged by this conviction, Dedekind, Cantor, and Weierstrass, by different methods but with a common aim, returned to the prob- lem of Eudoxus and stripped it of its disguised intuitive geom- etry. ‘Magnitudes,’ as we have seen, were replaced by ‘numbers,’ and geometrical intuitions were driven out to make room for those of traditional logic. Nebulous ‘quantities’ persisted in the analysis of some. In the numerical epsilons and deltas of rigorous Weierstrassian analysis the calculus of the nineteenth century attained its classic perfection. The e, 5 technique became part of the standard equipment of every working analyst, and an advanced course in the calculus toward the end of the century usually included the rudiments of Cantor’s theory of sets. Finality in 1900 In the preceding chapter we followed the development of the real number system to the close of the nineteenth century, and we have just seen that one origin of the modern concept of real numbers was analytic necessity. In the retreat of geometric and kinematic intuition to the classical logic which validated the work of Dedekind, Weierstrass, and Cantor, the calculus returned at the close of the nineteenth century to the paradoxes of the infinite that had exercised generation after generation of logicians from Zeno to Russell. Before further progress was possible, a more subtle logical technique had to be developed in the twentieth century, and this was forthcoming only when the FROM INTUITION TO ABSOLUTE RIGOR 295 symbolic logic prophesied by Leibniz was extended and refined far beyond his utmost imaginings. Thus, after two centuries, the calculus returned for new strength and health to one of the minds from which it had sprung. What it received will be our concern after we have reviewed some of the triumphs of analysis in the two centuries following Newton and Leibniz. For the moment, we recall the benediction pronounced by Henri Poincare (1854—1912, French) at the second international congress of mathematicians in 1900. On this historic and some- what solemn occasion, Poincare, the outstanding mathematician of his epoch and the Lagrange of the nineteenth century, con- trasted the roles of intuition and logic in mathematics. In par- ticular he reviewed the movement which has just been sketched and which, in the late nineteenth century, was called the arith- metization of analysis. The comforting assurances of this bold master of analysis induced a warm glow of security and pride in all who heard him, or who read his memorable address, and who, at least temporarily, had forgotten all they knew of mathe- matical history. Having recalled 53 that mathematicians had once been con- tent with the ill-defined and rough images of things mathe- matical as they appear to the senses or the imagination, Poin- care credited the logicians, for whom he had a dislike sharpening occasionally into acid ridicule, with having remedied this un- satisfactory state of affairs. Likewise, he continued, for irra- tional numbers and “the vague idea of continuity that we owe to intuition,” now (1900) resolved into “a complicated system of inequalities concerning integers.” By such means, he declared, all difficulties concerning limits and infinitesimals had been clarified. Today [1900) there remain in analysis only integers and finite or infinite systems of integers, inter-related by a net of relations of equality or inequality. Mathematics, as we say, has been arithmetined. ... Is this evolution ended? Have v.-c at last attained absolute rigor? At each stage of the evolution our fathers believed that they too had attained it. If they deceived themselves, do not we deceive ourselves as they did? We believe that sve no longer appeal to intuition in our reasoning. The philosophers tell us that this is an illusion . . . Now, in analysis today, if sve care to take the pains to be rigorous, there are only syllogisms or appeals to the intuition of pure number that could possibly deceive us. We may say today (1900) that absolute rigor has been attained. Here wc may refer to the last section of the preceding chapter. Some of the concrete achievements of analysis in applied mathe- matics will be discussed later. CHAPTER 14 Rational Arithmetic after Fermat We shall conclude our account of number since the seven- teenth century with a few typical items from the vast domain of the classical theory of numbers. Arithmetic in the tradition of Fermat, Euler, Lagrange, Legendre, and Gauss has been con- cerned mainly with the rational integers 0, ±1, + 2, . . . . Although it has attracted several of the greatest mathematicians since the seventeenth century, rational arithmetic has had far less influence than its nineteenth-century offshoot, the theory of algebraic numbers, on the rest of mathematics. Intensively cultivated for its own fascinations by hundreds of mathemati- cians of very different tastes, rational arithmetic has developed into an ever-growing expanse of loosely coordinated results with fewer general methods than any other major division of modern mathematics. From all this heterogeneous miscellany we shall select only three topics in which there is some coherence of method and an approach to completeness in certain details. The rest is largely a wilderness of dislocated facts offering a strange and discon- certing contrast to the modernized generality of algebra, geo- metry, and analysis. Much of it is hopelessly archaic in both aim and results. Rational arithmetic appears to be the one remaining major department of mathematics where generalizing a problem makes it harder instead of easier. Consequently it has attracted fewer merely able young mathematicians than any other. _ _ The subject falls naturally into the complementary divisions of multiplicative and additive arithmetic. The multiplicative theory develops the consequences of unique factorization into primes; the additive division is concerned with the composition 296 RATIONAL ARITHMETIC AFTER FERMAT 297 of integers as sums of prescribed types. A capital project in both divisions is that of enumeration: how many integers of a speci- fied kind satisfy given conditions? For example, how many primes are there between given limits? Or in how many ways may any integer be represented as the sum of a fixed number of positive cubes ? A problem in rational arithmetic is said to have been solved when a process is described whereby the required information is obtainable by a finite number of non-tentative operations. Time certainly is not the essence of the contract between the rational integers and the human intellect. The problem of resolving a number into its prime factors is solvable; yet the finite number of operations at present required for a number of a few thousand digits might consume more ages than our race is likely to have at its disposal. The problem of finding the prime factors of a number must strike an amateur as a natural one. To say that it has been solved in any respect that would satisfy common sense is a flattering exaggeration, and the like is true of many other arithmetical problems that seem natural to the inexperienced. The professional ignores these natural problems in favor of others which he or his predecessors have constructed, and for which he may hope to find at least partial solutions. Complete solutions, even of manufactured problems, are comparatively rare; and it would seem that rational arithmetic in the twen- tieth century is still relatively in the same position as geometry was before Descartes. Compared with what we should like to know in each of several directions, such progress as has been made is almost negligible. Yet all the resources of algebra and analysis have been hurled into the assault on this most elemen- tary of all divisions of mathematics. Outgrowths of diophantine analysis The nature of diophantine analysis has already been described in connection with Fermat. Its most extensive out- growth, the arithmetical theory of quadratic forms, slowly took shape during the eighteenth century, principally in the prodigi- ous output of Euler and the more restrained contributions of Lagrange and Legendre. Finally, in 1801, with the publication by Gauss of his Disquisitiov.es arithmeticae, diophantine analysis 1 in the sense of Fermat and Euler suffered an eclipse that was to last a century, until arithmeticians began to realize that the 298 THE DEVELOPMENT OF MATHEMATICS Gaussian theory of quadratic forms does not exhaust the subject of indeterminate equations. The second great branch of modern arithmetic that sprang from diophantine analysis was the theory of congruences. This also originated in the Disquisitiones. The suggestiveness of Gaussian congruence for modern algebra and the development of structural theories was noted in an earlier chapter. After Diophantus and Fermat, Euler was the great master of indeterminate analysis. But, like nearly all the predecessors of Lagrange, Euler contented himself with special sets of integers .or of rational numbers satisfying his equations. The only interest (such work has had at any time since Lagrange’s discussion of x z — Ay 2 = 1 in 1766-9 is in showing that a particular equation, or set of equations, with integer coefficients is in fact rationally or integrally solvable when the existence of a solution has been doubted. Thus a single numerical instance would dispose of the doubt (1945) concerning the solvability of a : 4 + y 4 + z 4 = w*, xyzzu 0, in integers. In a modern setting, this problem is equivalent to determining the number of representations of zero in the quartic form x i + y 4 + z 4 — w 4 and, if there are any, finding all. Euler (1772) conjectured that there are no solutions. The daring of this baseless conjecture typifies the cardinal distinction between indeterminate analysis before Lagrange and after him. Euler and others in the older tradition did not hesi- tate to suggest problems of great difficulty without offering the slightest suggestion for a method of attack. And when ingenuity furnished special solutions of an equation, the solver dropped the matter. Lagrange was the first to impose some common mathe- matical morality on diophantine analysis. He refrained from facile guessing; and when he did propose a problem, he also invented methods for obtaining its solutions. The turning point is marked by Fermat’s equation 2 x 2 — Ay 2 = 1, where A is any positive non-square integer, and all integer solu- tions x, y are sought. Fermat (1657) asserted that there are an infinity of solutions, a fact which Lord Brouncker and J. Wallis were unable to prove, although they gave a tentative method, improved in 1765 by Euler, for finding solutions. Euler proceeded from the conversion of "SjA into a continued fraction. But he was unable to prove the existence of a solution with y^O. Lagrange (1766-9) supplied the crucial proof, and in 1769-70, RATIONAL ARITHMETIC AFTER FERMAT 2 99 gave a non-tentative method for obtaining all integer solutions of x i _ Ay" 1 = B, where A, B are an y given integers. It was noted in earlier chapters that the Pythagoreans approximated to quadratic irrationalities by what amounts to solving special eases of Fermat’s equation by continued frac- tions, and that Brahmagupta in the seventh century gave a ten- tative method for solving x- — Ay~ + B in integers. But the mathematical distance between such empirical work as this and Lagrange’s proofs of necessity and sufficiency is immeasurable; and it is fantastic to claim that the Hindu mathematicians anticipated Lagrange. There is honor enough for Brahmagupta and Bhaskara in having imagined a problem that, centuries after they were dead, was to prove of cardinal importance in modern arithmetic. But in this they may have been merely lucky, for they devoted much time to numerous other problems that are essentially trivial. Fermat’s equation and its solution by Lagrange are indispensable in the Gaussian theory of binary quadratic forms, also in that of algebraic number fields of the second degree. Lagrange’s solution tvas the first determination of the units in an algebraic number field other than the rational. : In addition to being haphazard, Euler’s attack on diophan- tine equations was absurdly ambitious. If a single equation of the second degree in two unknowns proved unexciting, Euler in- creased the degree to three or four. If this failed to provide an attractive equation, he simultaneously increased the number of unknowns. As a last resort, he increased the number of equations and exercised his uncqualcd ingenuity on simultaneous systems. It is not surprising that he made but little progress toward either general methods or general theorems. Nor did any of his hun- dreds of successors who equaled or excelled him in ambition, but who fell far short of him in ingenuity. Advances toward real mathematics began when unambitious men like Lagrange and A. M. Legendre (1752-1833, French) confined their main efforts to the humble task of systematically investigating a single equation of the second degree in not more than three unknowns. Their work smoothed the way for Gauss, who also set himself a program which, compared to the rank opulence of the pre-Lagrangian period, is poverty itself. And without the pioneering work of Lagrange and Legendre, it is at least doubtful whether even Gauss would have been able to compose the Disquisitionrs. 300 THE DEVELOPMENT OF MATHEMATICS Arithmetical forms The basic technique of the arithmetic of forms originated with Lagrange’s theory of binary quadratics, in 1773, four years before Gauss was born. To describe it we shall use the standard terminology introduced in 1801 by Gauss and modified by later arithmeticians. Several of the definitions given next will be useful in slightly modified shape when we come to invariance. A form in rational arithmetic is a homogeneous polynomial P, P(x i, . . . , x n ), in the n indeterminates (or variables) Xi, , x n , with integer coefficients. If the degree is m, the form is called an 7t-ary m-ic. For n = 2, 3, 4, 5, . . . the forms are called binary, ternary, quaternary, quinary, . . . , respec- tively. In what follows, ‘form,’ unqualified, shall mean an «-ary m- ic. The fundamental concepts are equivalence and reduction of forms, and representation by a form. The form P{x j, . . . , x n ) is said to contain the form Q(4, . ■ ■ , x'„) if Q is derived from P by a linear homogeneous substitution T :Xi = a i \X l + • • • + ai n x'„{i = 1, . . . , n) with integer co- efficients aij whose determinant \a l3 \ is not zero. If \a l3 \ = ±1 the inverse, T~\ of T, expressing x[, . . . , x n as linear homo- geneous functions of Xi, . . . , x n , will have integer coefficients, and Q will contain P. When each of two forms thus contains the other, the forms are said to be equivalent; the equivalence of P, Q is written P ~ Q. It follows readily that this ~ is an instance of the abstract equivalence described in connection with Gaus- sian congruence. For P is either equivalent or not equivalent to Q; P ~ P; and if P ~ Q, then Q ~ P; also, P ~ Q and Q ~ R together imply P ~ R. Hence all forms equivalent to a given form are equivalent to one another; and therefore all forms may be separated into classes with respect to equivalence, two forms being put into the same class if and only if they are equivalent. The link with diophantine analysis is supplied by the concept of representation: an integer r is said to be represented by (or in) the form P{x i, . . . , x„) if and only if the equation " P(x 1, . * * , ^n) r is solvable in integers Xi, , x n . If Xi = s i, . . • ) = /« such a solution, (s i, ...,/„) is called a representation of r m P. The diophantine problem, as reformulated in 1773 by Lagrange, RATIONAL ARITHMETIC AFTER FERMAT 301 is to decide whether or not a given r is represented in P and, if it is, to find all representations. It follows at once from the definitions that if r is represented by a particular form in a given class, it is represented by every form in that class; and that if it is not represented by a particular form in the class, it is represented by no form in the class. The diophantine problem of finding all integer solutions of . . . , .v n ) — t is thus reduced to two others: to assign criteria, expressed in terms of the given coefficients of P, sufficient to decide whether or not r is represented by P; to find all the forms equivalent to P. The second of these suggests as a preliminary a third: given the coefficients of two forms, to determine whether the forms are equivalent and, if so, to transform one into the other. This in turn requires the automorphs of a given form, namely, those transformations which leave a form unaltered. Once the auto- morphs of P and one transformation taking P into Q arc known, all such transformations are known. For binary quadratic forms, the automorphs are obtained by solving certain of Lagrange’s equations x~ — Ay- — B. The remaining problem of modernized diophantine analysis, that of the reduction of forms, is on a different level. Suppose that in each class of forms it is possible to isolate a unique form by imposing appropriate conditions on the coefficients of all the forms in the class. Then this so-called reduced form, being equivalent to ever}' form in the class, may be taken as a representative of its entire class in the problems of equivalence and representation of numbers. Thus attention may be concen- trated on individual forms instead of being dispersed over possi- ble infinities of forms in the different classes. Incidentally, the problem of determining the number of classes of forms whose invariants have any preassigned integer values is suggested. Lagrange solved the problem of reduction for binary quadratics in 1773; a solution for ternary quadratics was first obtained by L. A. Sccbcr (German) in 1831. A little trial and a great deal of error will readily convince any experimenter that a complete solution of these basic problems is not to be anticipated in the immediate future. Nevertheless, their mere formulation was a notable achievement. If nothing else, they stripped the ancient diophantine analysis of its specious simplicity and revealed the nature of its inherent diffi- 302 THE DEVELOPMENT OF MATHEMATICS culties. In this respect they are an outstanding example of mathematical strategy as practiced by masters. These modern, clearly defined problems may prove to be so intractable in the general case that they will be abandoned. The entire program of a frontal attack on diophantine analysis has been questioned. Our successors may be forced to resume the classical technique of manufacturing problems which they can solve. We recall that diophantine analysis originated in the Pythagorean equation x 1 + y 2 = z 2 ; and it is conceivable that the generalized problem which sprang from this equation is artificial. The Pythagorean equation entered mathematics through geom- etry, not through arithmetic. A generation less respectful of tradition than ours may succeed in formulating and solving problems closer in some as yet unimagined sense to the nature of rational arithmetic, whatever that may be. In any event, interest in the modernized problems of diophantine analysis described above declined rapidly toward the close of the nine- teenth century. The problems were simply too hard; and all the impressive machinery of modernized algebra and analysis succeeded only in making a great clatter which failed to silence the insistent questionings of arithmetic. By far the major part of all the advances was in the theory of quadratic forms. A rough estimate of the amount of work done on the several kinds of forms gives eighty per cent in quadratic and twenty per cent in all others. Of the work on quadratics, about eighty per cent was devoted to binaries, eight per cent to ternaries, three per cent to quaternaries, and three per cent to n-aries. The remaining six per cent on quadratics was accounted for by binaries with coefficients in a few special quadratic fields. These statistics suggest that the general pro- gram of an arithmetical theory of forms was still largely a hope after over a century and a half of industrious exploitation by several hundred arithmeticians, including such men as Lagrange, Legendre, Gauss, Eisenstein, Dirichlet, Hermite, H. J. S. Smith, Minkowski, and Siegel. We shall now indicate briefly a few of the outstanding land- marks in the theory of forms of low degree. By his general treatment of binary quadratics (1773), Lagrange obtained incidentally and uniformly many of the special results of his predecessors, such as Euler’s theorem (1761) that every prime 6n + 1 is represented by x 2 + 3y 2 . Lagrange’s principal achieve- RATIONAL ARITHMETIC AFTER FERMAT 303 ment, however, was the introduction of universally applicable methods into the theory of binary quadratic forms. Legendre in 1798 published his Thcorie des nombres , the first treatise devoted exclusively to the higher arithmetic, in which Lagrange’s theory was simplified and extended. This work con- tains the earliest systematic attack on ternary quadratics. Much use was made of the law of quadratic reciprocity, of which the first complete proof was published by Gauss in 1801. With the Disquisitianes arillimeiicae (1801), the theory of binary quadratics crystallized into its classic shape. Systematiz- ing and completing details in the work of his predecessors, Gauss also added many new ideas of his orvn. Among the innova- tions was one which was to prove most unfortunate: Gauss con- structed his entire theory on forms ax- -f- 2 bxy -J- cy- with a, b, c integers. The even middle coefficient 2b makes the accom- panying algebra more elegant but needlessly complicates the arithmetic and leads to cumbersome refinements in classifica- tion. To an algebraist this may seem a trivial detail. But a moment’s reflection will show that as the subject under investi- gation is rational arithmetic and not algebra, the insistence that the middle coefficient be even is likely to cause unavoidable complications. Modern practice for binary quadratics (and to a lesser extent for ternaries), following Ivronecker, has returned to the un- restricted integer coefficients of Lagrange. Consequently it is necessary' to retain two vocabularies and to know which is being used in referring to papers on the subject. The first man to master the synthetic presentation of Gauss was Dirichlct, v'ho in 1863 summarized his personal studies and his recasting of the Disquisiliones in his Zahlenthcoric. The successive editions (1871, 1879, 1893) of this text 2 and Dirichlet’s earlier original contributions made the classical arithmetic of Gauss accessible to all without undue labor. A more significant advance of Dirichict’s in arithmetic generally will be noted later in connection with the analytic theory. Up to 1847 the arithmetical theory' of quadratic forms had been confined to binaries and ternaries. It might be thought that the extension to quadratics in 4, 5, 6, . . . indeterminates would be a matter of simple routine, like the passage from three dimensions to r. in analytic geometry. Hard experience quickly corrects this misapprehension; the difficulties of a detailed 304 THE DEVELOPMENT OF MATHEMATICS investigation increase rapidly with the number of indeterminates and even necessitate the invention of new principles. The first significant departure from the tradition of binaries and ternaries was F. M. G. Eisenstein’s (1823-52, German) arithmetical determination in 1847 of the number of representa- tions of an integer as a sum of six or eight squares. This was followed in 1847 and 1850 by an arithmetical determination of the number of representations of an integer without square factors as a sum of five or seven squares. In all cases only results were indicated, with no hint of the methods used. There is no doubt, however, that Eisenstein’s procedure was purely arith- metical and not analytic. Although all his results have long been details in the general theory of n-ary quadratic forms, they are of more than casual interest historically, as it was partly owing to them that the arithmetical theory was created. To trace the development, we must return to the determina- tion of the number of representations of an integer as a sum of two squares by Legendre in 1798 and, more simply, by Gauss in 1801, and to Euler’s unsuccessful struggle for forty years to prove that every positive integer is a sum of four integer squares, Euler’s failure was the steppingstone to Lagrange’s success in 1772, and thence to his own a year later. But neither obtained the number of representations. Quite unexpectedly the required number dropped out as an unsought by-product of an identity in elliptic theta constants, which Jacobi encountered in 1828 while developing the theory of elliptic functions. The like results for 2, 6, 8 squares are evident from other formulas in Jacobi’s Fundamenta nova of 1829. Those for an odd number of squares, lying much deeper, do not follow from similar identities. In passing, the problem of three squares was a famous crux in the arithmetical theory of quadratic forms until Legendre in 1798 published the first proof that all positive integers except those of the form 4 A (8& + 7) are sums of three integer squares. From all this it is clear that Eisenstein made a significant advance when he obtained his results for five and seven squares arithmetically. Possibly it was this work that moved Gauss to assert that “There have been only three epoch-making mathe- maticians, Archimedes, Newton, and Eisenstein.” If Gauss ever did say this (it is merely attributed to him), it is the most astounding statement in the history of mathematics. But as he may have said it, and as anything Gauss said about mathematics is to be taken seriously, we may briefly examine its tenabihty. RATIONAL ARITHMETIC AFTER FERMAT 305 Like Abel and Galois, Eisenstein was an “inheritor of unful- filled renown,” also of poverty and ill-health, and there is no guessing what he might have accomplished had he lived. But he enjoyed (after a fashion) about two years more of life than Abel, and eight more than Galois. His principal achievement outside of arithmetic was in elliptic functions, where he partly anticipated certain details of the Weierstrassian theory. His own analysis halted under the fatal disability of conditional convergence. 4 On the other hand, his applications of elliptic functions to the laws of cubic and biquadratic reciprocity were among the least ex- pected things in arithmetic. Against Gauss’ assertion are the facts that for one reference in living mathematics to Eisenstein, there are hundreds to Abel and Galois, to say nothing of Rie- mann and Dedekind, who were pupils of Gauss, or Eisenstcin’s less famous contemporary Rummer. Even in the narrowly limited domain of arithmetic, Eisenstein’s influence has been slight in comparison with that of Dedekind. In the older form, now obsolete, of the theory of higher reciprocity laws, Eisen- stein’s work of 1850 was vital; but here again the generative concept of prime ideal divisors was Rummer’s. It seems unlikely, then, that posterity will revise the almost universal verdict of 1945, that up till then the three epoch-making mathematicians were Archimedes, Newton, and Gauss. The principal steps toward a general theory of n-ary quad- ratics appear to have been as follows. Perfecting and greatly extending the theory for « — 3 as left by Gauss in the Disquisi- tiones, Eisenstein in 1847 introduced new principles for the classification of ternaries into orders and genera. Hcrmitc in 1850 simplified the theory of reduction for ternaries, and in 1S51 devised his general analytic method of continual reduction. The theory of ternaries was further developed by Eisenstein in 1851-2; by H. J. S. Smith (1S26-1S83, Irish) in 1867; by E. Selling (German) in 1874, and by many others in the lS50’s-70’s. In 1864 and again in 1S67, Smith initiated one form of the general theory of r .- ary quadratics from which Eisenstcin’s theorems on five and seven squares were easily obtainable. Owing partly to the conciseness of the exposition, these and other detailed consequences were overlooked, and the problem of five squares was proposed by the French Academy for its Grand Prix in 1SS2. Brevity in mathematics is sometimes the soul of obscur- ity. Smith elaborated the relevant parts of his general theory of 1S64-7, and shortly after his death shared the prize with H. 306 THE DEVELOPMENT OF MATHEMATICS Minkowski (1864-1909, Russian; Germany) then a student of eighteen at the beginning of his too brief career. Thus, after an unnecessary delay of over a quarter of a century, the general arithmetical theory of n-ary quadratics was launched with complete eclat. Poincare, Minkowski, and others further developed the theory in the two succeeding decades. With the exception to be noted immediately, little that could be considered basically new w r as done till C. L. Siegel (German) in 1935 gave a profound reworking of the entire theory. The new acquisition, that of the geometry of numbers, was created almost entirely by Minkowski, although special in- stances 3 of it occur in the early (posthumously published) work of Gauss; in a project of Eisenstein’s (1844); in Dirichlet’s (1849) asymptotic evaluations of sums of arithmetical functions; in the work already cited on the reduction of ternary quadratics; and in the semi-geometrical presentation of the theory of elliptic modular functions by H. J. S. Smith in 1876, and the similar revision of binary quadratics by Poincare in 1880. One basic principle is so simple as to appear ridiculous: if n -p 1 things are stored in n boxes, and no box is empty, exactly one of the boxes must contain two things. The solution of a trick problem popular some years ago follows from this principle of geometrized arithmetic: state necessary and sufficient condi- tions that there shall be at least two human beings in the world with the same number of hairs on their heads. The first published results in the geometry of numbers appear to be Eisenstein’s geometrical proof (1844) of the Gauss lemma for the proof of the law of quadratic reciprocity, and his formula (1844) for the number of solutions of x 2 -f- y 2 ^ n in integers x, y, where n is given. A lattice point being defined as a point whose coordinates are integers, the last is equivalent to finding the number of lattice points contained by a circle, including its circumference, with center at the origin and radius Min- kowski developed meager hints like these into a powerful method which was applied with conspicuous success by himself and many others to difficult questions in the theories of forms, especially linear forms with real coefficients, and algebraic numbers. It is not necessary, of course, to restate an arithmetical problem geometrically; but doing so suggests to those with spacial intui- tion in n dimensions analytic processes which they might not imagine otherwise. With or without appeal to spacial imagery, RATIONAL ARITHMETIC AFTER FERMAT 307 the type of problem suggested by the geometry of numbers inspired much work in the analytic arithmetic developed since about 1910 by the schools of E. Landau (1877-1938, German), G. Ii. Hardy (18 77-, English), J. E. Littlewood (18SS-, English), and S. Ramanujan (1887-1920, Indian) in England. Arithmetic thus repaid its heavy debt to analysis by showering some of the foremost classical analysts of the twentieth century with an abundance of difficult problems. The arithmetical theory of forms of degree higher than the second was responsible for much less. Eisenstcin initiated (1844) the theory of binary cubics, and in so doing came upon the first algebraic covariant in history. But he did not exploit his dis- covery, although he realized its suggcstivencss. The arithmetic of binary cubics was reworked by the British mathematicians G. B. Mathews and W. E. H. Berwick in 1912. Progress up to 1945 in the arithmetic of forms beyond this point, with three exceptions, was inconsiderable. The norm of an algebraic integer is the product of all its conjugates; the norm equated to unity defines the units of the field concerned. Dirichlet (1840) proved the basic theorems for such units, al- though there is yet no practicable way of obtaining them even in special cases for fields of degree higher than the third. This work generalized Lagrange’s on Fermat’s equation. Dirichlct’s prob- lem of units and its immediate extension to the representation of any number by a general norm are the most immediate generali- zation of the theory of binary' quadratics. Today they are a topic in algebraic numbers. The origin of this farthest outpost in the systematized theory of forms was Lagrange’s remark (1767) that the norm of a general algebraic number repeats under multiplica- tion, and this in turn can be traced back to Fibonacci’s identity. It may interest some to know that Diricblet’s inspiration came to him in church while he was listening to the music on an Easter Sunday. The second exception to the general rule of sterility beyond real quadratics was Hcrmitc’s introduction in 1854 and 1857 of the forms since known by his name. In the binary case, a Hcrmitian form is of the type ax: v' -f- bxy’ -f b’x'y -f- cry', where a, c are real constants, b , V conjugate imaginary constants, and the variables in the pairs a\ x' and y, y f arc conjugate imaginarics, so that the entire form is real and hence capable of representing real numbers. From Ilermite’s arithmetical theory' of these forms in two or more variables evolved the extensive theory of 308 THE DEVELOPMENT OF MATHEMATICS Hermitian forms and matrices, which after 1925 became familiar to physicists through the revised quantum theory. Hermite also ( 1849 ) initiated the closely related arithmetical theory of bilinear forms, 6 thus starting much algebra that is now standard in a college course, including parts of the theory of matrices and elementary divisors. The last originated explicitly in H. J. S. Smith’s discussion of systems of linear diophantine equations and congruences (1861), and was developed independently by Wcierstrass and by G. Frobenius (1849-1917, German) in the 1870’s-80’s. The point of historical interest here is that all these extremely useful techniques of modern algebra, which after 1925 became commonplaces in mathematical physics, evolved from quite useless problems in the theory of numbers. The third and last exception to general sterility connects the arithmetic of forms with that other major outgrowth of ancient diophantine analysis, the Gaussian concept of con- gruence. Dickson in 1907 began the congruencial theory of forms, in which the coefficients of the forms are either natural integers reduced modulo p, p prime, or elements of a Galois field. The linear transformations in the theory, corresponding to those in the classical problem of equivalence, were similarly reduced, and hence modular invariants and covariants were definable. By 1923 the theory was practically worked out, except for two central difficulties, by Dickson and his pupils. Simplified derivations for some of the results were given (1926) by E. Noether by an application of her methods in abstract algebra. Before passing on to congruences, we note an outstanding advance in the older tradition of diophantine analysis. If /(z) = a n z n + a n _iz n_1 + • • • + a# + a 0 is an irreducible polynomial of degree n ^ 3 with integer coeffi- cients, and if H(x, y ) s a n x n + tfn-i* n-3 y + * * * + dixy"” 1 + a 0 y n is the corresponding homogeneous polynomial, then H(x, y) = where c is an integer, has either no solution or only a finite num- ber of solutions in integers x, y. This is the capital theorem (1909) of A. Thue (Scandinavian). A generalization 7 was given by Thue himself, and another (1921) by Siegel. After all that has been said aboup the paucity of general methods and the plethora o fragment.* ^ results in diophantine analysis, Thue s theorem speaks forltself. It was proved by elementary methods. RATIONAL ARITHMETIC AFTER FERMAT 309 The theory of congruences The subject of congruences in rational arithmetic is usuallv assigned to the multiplicative division, although it is concerned chiefly with the detailed investigation of one highly specialized type of diophantine equation. a T x~ -f - • • -f a ; .v tCo= my, in which*, y are theindeterminates. and the coefficients a n , . . . , a u o D , vi are given constant integers, with a n ^ 0, m 0. The essential point is that one of the indeterminates, y, occurs only to the first degree. The cases v: = ±1, being of no interest, are excluded. The equation rewritten as a congruence is o„x Tt -f- • * * -r oi.x -f Co s 0 mod m ; m is called the modulus; ?: is the degree; and solving the dio- phantine equation is equivalent to finding all integer values of x, called the roots of the congruence, that make the polynomial on the left a multiple of m. If x = c is a solution, so also is c -f km, where k is any integer. Since c 4- km s c mod in, it suffices to find all solutions whose absolute values do not exceed \rr\j2. These arc said to be incongruent modulo m. The statement of the generalized problem for one congruence in several indeter- minates, or for a simultaneous system of such, is immediate. We refer here to what has already been said about congruences in connection with algebraic structure. Gauss does not disclose 5 what led him to this cardinal con- cept of modern arithmetic. But its systematic use in the earlier sections of the Disquisiiiones enabled him to unify and extend important theorems of Fermat, Wilson, Euler, Lagrange, and Legendre on arithmetical divisibility and, in the famous seventh (concluding) section, to give a reasonably complete theory of binomial equations in algebra. A few of the older results on divisi- bility maybe restated in the language of congruences to illustrate the general ideas. If p is prime, ar* -1 — 1=0 mod p, has exactly p — 1 incon- gruent roots. This is Fermat’s theorem, proved essentially by the implied method of congruences by Lagrange (1771), who showed also (176S) that a congruence of degree r. has not more than v. incongruent roots for a prime modulus. The theory of residues of powers, originating with Euler in 1769, is concerned with the general binomial congruence x n - enfl mod m. It has numerous applications in algebra, particularly in the theory of equations and in finite groups. One crucial unsolved problem 310 THE DEVELOPMENT OF MATHEMATICS may be noted. If p is prime, r any number not divisible by p and if p — 1 is the least value of n for which r n — 1=0 mod p, r is called a primitive root of p. A prime p always has exactly 4>{p — 1) primitive roots, where p{n) is Euler’s function denoting the number of positive integers not greater than n and prime to n. The problem is to devise a practicable, non-tentative method for finding at least one primitive root of any given prime. Between Euler’s initiation of the subject in 1769 and 1919 — a century and a half — 232 lengthy articles and short notes were published on binomial congruences. None made any substantial progress toward a solution of this crucial problem. Another outgrowth of Euler’s power-residues is the entire theory of reciprocity laws, already noted in connection with modern algebra. Yet another is the extensive theory of functions defined for integer values of their variables, all of which evolved from the theorem of Gauss that 2<£(d) — n i where the summa- tion extends to all divisors d of the fixed integer n, and p is Euler’s function. Practically every arithmetician of note and a host of humbler workers have contributed to the theory of congruences since Gauss started the subject in 1801. For all their efforts, two cen- tral problems of the theory defy solution: to assign criteria on the given coefficients of a system (one or more) of congruences to decide whether or not the system is solvable and, if it is, to find all its incongruent solutions non-tentatively. For a single con- gruence of the first degree in any number of unknowns, the problem is more completely solved than are most of the solved problems in arithmetic, and likewise for a simultaneous system of such congruences, the solution having been given (1861) by H. J. S. Smith. The higher reciprocity laws represent the farthest advance in the case of binomial congruences; their complexity may hint that the general problem is intractable by methods known up to 1945. It must be remembered that these problems and others like them in classical arithmetic have not been the easy sport of mediocre men; some of the most powerful mathe- maticians in history have wrestled with them. Congruences were responsible for one theory of far more than merely arithmetical interest. The notation for a congruence suggests the introduction of appropriate ‘imaginaries to supply the congruence with roots equal in number to the degree of the congruence when there is a deficiency of real roots. As in t e corresponding algebraic problem, it is not obvious that xmagi RATIONAL ARITHMETIC AFTER FERMAT 311 narien can be introduced consistently. That they can, was first proved in 1830 by Galois, who invented the required ‘numbers,’ since called Galois imaginaries, for the solution of any irreducible congruence Fix) — 0 mod p, where p is prime. He thus obtained a generalization of Fermat’s theorem, and laid the foundation of the theory of finite fields. As remarked by Dickson, 5 “Galois’s introduction of imaginary" roots of congruences has not only led to an important extension of the theory of numbers, but has given rise to wide generalizations of theorems which had been obtained in subjects like linear congruence groups by' applying the ordinary theory of numbers.” Galois was eighteen when he invented his imaginaries. We pass on to the third and last division of rational arith- metic which we shall consider. Here great progress has been made, most of it since 1895. Applications of analysis Since the time of Euler, analysis has been applied incidentally to rational arithmetic; but it was only in 1839, with Diri- chlct’s Recherches sur diverses applications dc 1' analyse infinitesi- ntale a la iheorie dcs nombres , that limiting processes entered organically into the theory of numbers. Before Diricblet, such analysis as was used remained in the background , y ‘ arithmetical results being obtained by' the device of comparing coefficients in two or more expansions of a given function by different algo- rithms. This technique originated with Euler in his work of 1 7*1 8 in the theory of partitions, a subject which he initiated in 1741. After Dirichlct, the next organic uses of analysis were Hermite’s (1851) in his method of continual reduction, and Ricmann’s (1859) in the distribution of primes. But it was not until the twentieth century' that modern analysis was sys- tematically ' applied to additive arithmetic. Simultaneously' there was an unprecedented advance in the application of analysis to the multiplicative division. We shall presently' describe only enough to illustrate the radical difference between the old and the new. With the exception of a very' few outstanding acquisi- tions like Thue’s theorem, it seems probable that the early twentieth century will be remembered in the future history of rational arithmetic chiefly for its achievements in the analytic theory'. There remains, however, what some consider a desideratum: to obtain those results of the analytic theory' which do not 312 THE DEVELOPMENT OF MATHEMATICS involve a limiting process in their statement without an appeal to continuity. Thus Dirichlet proved analytically that there are an infinity of primes of the form an + b, where a, b are constant relatively prime positive integers and n runs through all posi- tive integers. Attempts by Emmy Noether and others in the 1930’s to obtain this non-analytic theorem without the use of analysis failed. On the other hand, all of Euler’s and Jacobi’s theorems on partitions, and all of the non-analytic theorems on numbers of representations in certain quadratic forms, first obtained analytically by Jacobi and others, have been proved without analysis. The reasons for failure in one instance and success in another superficially indistinguishable from it are not understood. It will be convenient to call demonstrably avoidable analysis inessential; and to speak of analysis which has not yet been proved inessential, or which leads to final results implying a use of continuity, as essential. Kronecker would probably not have admitted essential analysis into arithmetic, and might even have declared that its products are as inexistent as irrationals. A classic example of the use of analysis later seen to be inessential occurs in many of the applications of elliptic and modular functions to the Gaussian theory of binary quadratic forms. In the hands of Kronecker, Hermite, and a score of less notable mathematicians, a close connection between binary quadratics and the theory of complex multiplication of elliptic functions was developed after 1860 into an extensive department of arithmetic. One detail of all this intricate theory exemplifies the analytic peculiarities. From a passage in the Disquisitiones it appears that as early as 1801 Gauss had effected the difficult determination of the number of classes of binary quadratics having a given determinant. The first published determination was Dirichlet’s of 1839, in which analysis was essential. (An outstanding desideratum is a ‘finite’ proof of Dirichlet’s results.) For forms of a negative determinant, Kronecker in 1860 found several remarkable formulas whereby the number of classes can be calculated recursively without analysis. These formulas ap- peared as by-products of Kronecker’s investigations in elliptic functions, and were the heralds of several hundred by later writers, many of whom used elliptic modular functions to obtain their results. We shall not pursue this matter further here, as a detailed account belongs to the specialized theory of numbers and we can attend only to matters of more than local significance. RATIONAL ARITHMETIC AFTER FERMAT 313 The point of interest here is that whereas analysis was essential in Dirichlct’s derivation of the class-number, and might therefore have been reasonably expected to appear essentially in the deduction of the recurrence relations, it actually proved to be inessential for the latter- Arithmeticians who insist (there arc such) that a method containing analysis essentially 11 belongs to analysis and not to arithmetic would claim that Kroneckcr’s formulas, not Dirichlct’s, are the arithmetical solution of the class-number problem. There is more than a pedantic difference between the two opinions, at least historically. Experience has shown that the search for proofs and theorems independent of essential analysis frequently turns up unexpected simplicities and reveals new arithmetical phenomena. Gauss emphasized the desirability of multiplying proofs in arithmetic with a view to making the abstruse clear. However, arithmetic is sufficiently broad and difficult to permit all types of workers to follow their own inclinations. Beginning about 1917, the general trend was toward essential analysis. The theory of partitions illustrates the historical disconti- nuity between inessential and essential analysis. If P(n) denotes the total number of ways the positive integer n is obtainable as a sum of positive integers, it is obvious, as noted by Euler in 1748, that P(n) is the coefficient of x n in the expansion of [11(1 - * r )] 1 into a power series in x. With unsurpassed manipulative skill, Euler derived numerous identities between this infinite product and others suggested by problems in parti- tions, thus anticipating many formulas in elliptic theta constants dcduciblc from Jacobi’s presentation (1828-9) of elliptic func- tions. An extensive literature sprang from these discoveries. Much of it is algebraic, and in none is analysis essential. Indeed, Sylvester, no analyst, desiring to understand the subject and being too impatient to master elliptic functions, developed a hint thrown out by N. M. Ferrers (1S29-190S, English) in 1853 into a graphical theory in "which some of the properties of parti- tions can be inferred from point lattices. But although the pictorial representation may have enabled Sylvester and some others to avoid inessential analysis in their thinking, it added nothing new to the theory of partitions. Among other results of the Euler-Jacobi tradition were numerous elegant formulas whereby P(n) and other partition 314 THE DEVELOPMENT OF MATHEMATICS functions could be calculated recurrently. Each afforded a com- plete arithmetical solution of the problem of computing the function of n concerned for any value of n. That is, all were use- less as aids to practical computation for any but inconsiderably small numbers. The break came in 191 7, when G. H. Hardy (1877- English) and S. Ramanujan (1887-1920, Indian) applied their new analytic methods to the derivation of an asymptotic formula for P(n), which put the practical evaluation of P(n) into touch with common sense. Before this, it had required a month’s labor by a leading expert in the older methods to compute P(200); only six terms of the asymptotic formula gave P(200) = 3,972,999,029,388 with an error of .004. This detail typifies the computational superiority of formulas depending on essential analysis over the corresponding exact theorems preferred by pure arithmeticians. The analysis used is applicable to many other numerical func- tions appearing as coefficients in expansions of certain functions of a complex variable not continuable beyond the unit circle. This work of 1917, like most of the essential analysis in modem rational arithmetic, was of more than local interest in its own domain. It instigated a greatly increased activity in refined classical analysis and in the modem theory of inequalities. One of the most famous problems in all arithmetic, that of the distribution of primes, yielded to analysis only in 1896, when J. Hadamard 11 (1865—, French) and C. J. de la Vallee- Poussin (Belgian) proved independently that the number N(x) of primes sS x is asymptotically equal to ar/log x , that is, the limit of N{x) - [.r/log at ] -1 as x tends to infinity is 1. This is usually called the prime number theorem. The relevant history would fill a book. 13 Legendre, Gauss, and others had proposed formulas inferred from actual counts of primes; but it cannot be said that any of these tentative efforts materially furthered the final success. The great Russian mathematician, P. Tchebycheff (1821-1894) in 1830-1 made the first considerable advance since Euclid in the theory of primes; but much sharper analysis was needed than any available until the last decade of the nineteenth century. What appears likely to remain for some time the best proof of the theorem is Landau’s reformulation (1932) of that of N. Wiener (1894—, U.S.A.), who deduced the result almost as a corollary from his work on Tauberian theorems. The latter, so RATIONAL ARITHMETIC AFTER FERMAT 315 / named by Hard)* after the German analyst Tauber, evolved from the converse of Abel’s theorem on convergent power series. Even the briefest notice of the theory of primes must men- tion the famous conjecture known as Riemann’s hypothesis, which is to classical analysis what Fermat’s last theorem is to arithmetic. Euler (1737) noted the formula Stt - ’ = 11(1 — p ~- *)~ 1 , the sum extending to all positive integers n, and the product to all positive primes p. The necessary conditions of convergence hold for complex values of s with real part >1. Considering 2?z - * as a function f(r) of the complex variable s, Riemann (1859) proved that f(r) satisfies a functional equation involving £(j), f(l — s), and the gamma function of s. He was thus led to the theorem that all the zeros of f (r), except those at s = —2, —4, —6, . . . , lie in the strip of the r-planc (Argand diagram for s) for which 0 ^ cr < 1, where a is the real part of s. His theorems would be even more interesting than they are if all the zeros in the strip should lie on the line cr = 4-. Riemann conjec- tured 14 that this is so. Attempts to prove or disprove this con- jecture have generated a vast and intricate department of analysis, especially since Hardy proved (1914) that f(s) has an infinity of zeros on cr = h. Although the question was still open in 1945, scores of profound papers bristling with thorny analysis had enriched the literature of analytic arithmetic for almost a third of a century, some of them based, however, on the sup- position that Riemann’s conjecture is true. This bold technique of inference from doubtful conjectures was something of a new departure in arithmetic, where the tradition of Euclid, Lagrange, and Gauss had stickled for proof or nothing. The rationalized justification for the novel proce- dure, were any needed, was the unrealized hope that by trans- forming a dubious hypothesis into something new and strange, an accessible equivalent would sometime, somehow, drop out. Still on the assumption that Riemann's hypothesis and other unproved conjectures of a similar character are true, numerous profound theorems on the representation of numbers as sums of primes, or in other interesting forms, were skillfully deduced by some of the most refined analysis of the twentieth centur}'. Should any of these boldly conceived but unborn theorems ever materialize, they will be among the most remarkable in arithmetic. Adhering more strictly to the Euclidean tradition of proof before prophecy, the Russian mathematician I. M. Vinogradov, 316 THE DEVELOPMENT OF MATHEMATICS beginning about 1924, developed new methods in the analytic theory of numbers, and in 1937 apparently came within a reasonable distance of proving another famous guess concerning primes: every even number >2 is a sum of two primes. C. Gold- bach (1690-1764, Russia) in 1742 confided this conjecture to Euler who, while believing it to be true, confessed his inability to prove it. Anyone who has inspected the analysis by which Vinogradov proved (1937) that every odd number beyond a certain point is a sum of three odd primes will sympathize with Euler. The best previous result was that of L. Schnirelmann (Russian), who proved (1931) that there is a constant n such that every integer >0 is a sum of n or fewer primes. But the method of proof, according to Landau, was incapable of further refinement. Vinogradov’s theorem was conjecturally derived in 1923 by Hardy and Littlewood from an unproved mate of Riemann’s hypothesis. From 1896 till 1940 a major part of analytic arithmetic originated in the theory of rational primes. Some of this was extended to algebraic numbers, as when Landau (1903) obtained the prime ideal theorem corresponding to, and including, the prime number theorem. Here the necessary analysis proceeds from Dedekind’s generalization (1877) of Riemann’s £(s) to algebraic number fields. Another, less extensive, research gen- eralized Dirichlet’s work on the class-number of binary quad- ratics, being concerned with the number, proved to be finite, of distinct classes of (integral) ideals in an algebraic number field. The explicit determination of this number in an approach- able form is one of the unresolved cruxes in arithmetic. All of this work belongs to the multiplicative division. Equally prolific of new analysis and far-reaching theorems in arithmetic was the advance beginning in 1909 with Hilbert’s solution of Waring’s problem. The English algebraist E. Waring (1734— 1798) emitted the conjecture (1770) that every integer n > 0 is the sum of a fixed least number g(s) of sth powers of integers ^0. For s = 2, this is the result proved by Lagrange and Euler that every positive integer >0 is a sum of four integer squares ^0. Since no integer 4 A (8& + 1) is a sum of three squares, it follows that g( 2) = 4; and it is known that g(3) = 9. Waring himself proved no single case of his problem; nor did he offer any suggestion for its solution. For all. that he or anyone else in the eighteenth century knew, g(s) might not exist. RATIONAL ARITHMETIC AFTER FERMAT 317 It so happens, however, that Waring’s guess was one of those few in the theory of numbers that have started epochs in arithmetic. Little of any significance issued from Waring’s conjecture until about a century and a half after it had been made on only scanty numerical evidence. The theorem itself might be guessed after an hour’s figuring. For example, g{ 4) may be 19 as stated by Waring, a result which had not been proved as late as 1945. It used to be imagined by romanticists that Waring and other rash guessers in arithmetic knew mysterious methods, now lost, of extraordinary power. There is no evidence that they did not. But professionals who appreciate the inherent difficulties of arithmetic believe that the lost methods, with the possible excep- tion of Fermat’s, are mythical. Gauss, for example, wdicn urged in 1818 to compete for a prize offered by the French Academy for a proof or disproof of Fermat’s last theorem, expressed him- self quite forcibly on the undesirability of facile guessing in arithmetic. Including Fermat’s theorem in his remarks, Gauss declared that he himself could manufacture any number of such conjectures which neither he nor anyone else could settle. All questions of mathematical ethics aside, it is at least possible that stating difficult problems with no hint of a method for attacking them is more detrimental than advantageous to the progress of arithmetic. Unless we arc eager to believe that certain individuals are divinely inspired and can foresee what course mathematics should follow to accord with the inscrutable verities of Plato’s Eternal Geometer, we may suspect that base- less guessing is likely to deflect talented originality into artificial channels. Waring was an accomplished algebraist, but there is no evidence that he was inspired; and it seems like nothing but blind luck that his easy guess led to anything more profound than trivialities. That it did finally prove extremely stimulating may appear in the long run disastrously unfortunate. For there is little doubt that Waring's delayed success was largely responsi- ble for the return about 1920 to the prc-Lagrangian tactic of deduction based on conjectures. Of course Lagrange and Gauss may have been mistaken or merely pedantic in their rejection of published guessing as a stimulus to progress, and the ninc- tccnth-ccntury caution may have been excessive. If so, the early twentieth century will doubtless be long remembered as the dawn of a new era in arithmetic. 318 THE DEVELOPMENT OF MATHEMATICS Hilbert’s proof (1909) of Waring’s conjecture established the existence of g(s) for every s, but did not determine its numerical value for any s. The curiously ingenious proof, shortly simplified by several mathematicians, depended on an identity in 25-fold multiple integrals, and like much of its author’s mathematics aimed only at existence without construction. Its historical im- portance is less that it was the first solution of an outstanding problem than that it incited analysts to find at least a bound to the numerical value of g(s) for any given s. It was the latter problem and the cognate one to be described presently that were largely responsible for the explosive outburst of analytic arithmetic in the 1920’s-30’s. As already implied, this work marks an epoch in the theory of numbers. Hardy and Littlewood in 1920-8 invented the analytic method for Waring’s problem which was to remain the stand- ard till Vinogradov, having started in 1924 from methods similar to those of the English arithmeticians, developed his own more penetrating technique in the 1930’s. The problem affiliated with g(j-) is that of finding G(r), defined as the least integer n such that every positive integer beyond a certain finite value is the sum of n 5th powers of integers ^0. Thus the best value of g(4) up to 1933 was g(4) g 35, in contrast with Hardy and Little- wood’s G( 4) 19, while it was shown (1936) that G(4) is either 16 or 17. For s > 6, Vinogradov’s improved methods gave (1936) much smaller values of G(s ) than those obtained previously. Although the pioneering methods were thus superseded, their influence on the development of asymptotic analysis remained incalculable. Utilizing results of Vinogradov’s, Dickson and S. S. Pillai (1902-, Indian) in 1936 proved independently an explicit formula for g(s), valid for all s > 6, except possibly for certain doubtful cases. 15 It is gratifying to report that since the first edition (1940) of this book, these doubtful cases have been disposed of (1943) by I. M. Niven (1915-, U.S.A.). Thus, after 169 years Waring’s guess was finally proved. In addition to instigating a vast amount of acute analysis, the problem had suggested numerous others solvable by similar methods, such as the representability of all, or ‘almost all, positive integers as sums of polynomials taking integer values for integer variables, or as sums of squares and primes. As this is written, there is no sign of an abatement in the output of analyt- ic arithmetic. Two of the leading experts in the subject, Ramanujan and Landau, died before their time. The others RATIONAL ARITHMETIC AFTER FERMAT 319 more directly responsible for the creation of new methods were still active in 1945, and a crowd of younger men was coming on. Another isolated result solidified by modern analytic methods may be cited for its curious implications regarding the sup- pressed work of the initiator of the modern era in arithmetic. C. L. Siegel (1 896 — , Germany, U.S.A.) in 1944 gave the first proof of a statement (1801) by Gauss concerning a certain asymptotic mean value in the theory of the binary quadratic class number. As the relevant formula could hardly have been inferred from numerical examples, it would be interesting to know how Gauss satisfied himself of its correctness. In any event, it is indicative of the lawless difficulties of the theory of numbers that a result stated by Gauss should have stood in the classic literature for 143 years without proof. The 1920’s— 1 930’s witnessed the beginning of an era in arith- metic comparable to that inaugurated by Gauss in 1801. Analysis, the mathematics of continuity, had at last breached outstanding problems in the domain of the discrete. That explicit integer values for numerical functions like g(j), G(r) should be obtainable by analysis would have seemed miraculous to the arithmeticians of the nineteenth century. The like holds for the modern work in the theory of primes, and in other parts of multiplicative arithmetic. It is therefore not true, at least in the theory of numbers, that all the great mathematicians died before 1913. In this chapter, as in others, literally hundreds of worthy names have been passed over without mention, and likewise for dozens of extensive developments to which scores of workers in the past two centuries have devoted their lives. The topics described arc, however, a fair sample of some of the best that has been done in rational arithmetic since Fermat. CHAPTER 15 Contributions from Geometry With a literature much vaster than those of algebra and arithmetic combined, and at least as extensive as that of analysis, geometry is a richer treasure house of more interesting and half-forgotten things, which a hurried generation has no leisure to enjoy, than any other division of mathematics. Con- tinually changing ideals and objectives in the development of geometry since the seventeenth century have made it impossible for students and working mathematicians to be aware of hun- dreds of theorems, and even extensive theories, that the geom- eters of the late nineteenth century prized as objects of rare beauty. On a rather humble level, for example, it was held by compe- tent geometers in 1940 to be a sheer waste of effort for a student contemplating a career in geometry, or in any division of living science or mathematics, even to glance at the so-called modern geometry of the triangle and the circle, created largely since 1870. Yet it has been said, no doubt justly, that almost any theorem of this intricate and minutely detailed subject would have delighted the ancient Greeks. And that, precisely, is the point. All the classic Greek geometers were buried or cremated two thousand years ago. Geometry in the meantime has ad- vanced. By 1900 at the latest, special theorems in Euclidean geometry were no longer even a tertiary objective of creative geometers, no matter how beautiful or how interesting they might appear to their authors. This does not imply that such theorems were valueless to those who could appreciate them; they preserved more than one isolated teacher from premature fossilization. They may also have irritated some who later became skilled geometers into 320 CONTRIBUTIONS FROM GEOMETRY 321 finding out what modern geometry is about. On the other hand, many a working mathematician of the 1930’s looked back with something akin to rage in his heart on the months or years squandered on this sort of geometry, or on the interminable properties of conics, at the very time of his life when his capacity for learning was greatest and when he might have been master- ing some living mathematics. In defense of this waste, if such it were, it was argued that English schoolboys stili took a keen pleasure in these intriguing puzzles of their forefathers. No doubt they did. But the further claim that such a training made first-rate geometers is con- tradicted by the evidence. The attempted justification on the grounds of mental discipline may be left to the psychologists. In any event, it seemed slightly fatuous to impart discipline through outmoded fashions when so much of equal difficulty and vital necessity had to be mastered if one were to think geometrically in the manner demanded by a continually advanc- ing science and mathematics. The foregoing opinions, it may be stated, were those of three of England’s foremost mathemati- cians in the 1930’s, all of whom had made high marks in this prehistoric sort of geometry in competitive examinations for English university scholarships. However, even in the most elementary geometry an original and ingenious mind may occasionally think of something to do rather unlike what is already classical. The more orthodox Greek geometers, we saw, limited themselves to a straightedge and compass in their permissible constructions. Why not dis- pense with one or other of these traditional implements? It occurred to G. Mohr (Danish) in the seventeenth century to sec what constructions could be performed with a compass alone, and L. Maschcroni, (1750-1800, Italian) actually %vrote a book on the geometry of the compass. Napoleon Bonaparte is said to have been highly elated by his solution of a simple problem in Maschcroni’s geometry. Others hobbled themselves by using a straightedge alone, or this with one given point in the plane, and so on. Finally it occurred to E. Lemoinc (1840— 1912, French), to attempt to assign a measure of the com- plicatcdness of a geometrical construction. He presented an account of his proposals (18SS-18S9, 1892-1S93) at the Inter- national Mathematical Congress held in connection with the Chicago World Fair of 1893. He succeeded in defining the simplicity of a construction in terms of five operations of clc- 322 THE DEVELOPMENT OF MATHEMATICS mentar y geometry, such as placing one point of a compass on a given point; the simplicity -was the total number of times these operations were used. It was thus possible to assign marks to different constructions for the same figure, but a method for finding the construction with an irreducible number of marks seemed to be lacking. Once more it was demonstrated that the only royal road to elementary geometry is ingenuity. Another type of problem the beginner meets may have suggested the hotly controversial and partly discredited theory of enumerative geometry: how many lines, circles, etc., satisfy a prescribed set of conditions? Or, to take an instance from the 1940’s, what practical use, if any, can be made of Pascal’s theorem in conics? Almost exactly 300 years separate Pascal’s discovery of the theorem and its application by aeronautical engineers to lofting; naval architects might well have used it earlier. At a slightly more advanced stage, almost anyone can invent his own peculiar system of coordinates and proceed to elaborate the geometry it suggests. Many have. Rising to a considerably higher level of difficulty, we may instance what the physicist Maxwell called “Solomon’s seal in space of three dimensions,” the twenty-seven real or imaginary straight lines which lie wholly on the general cubic surface, and the forty-five triple tangent planes to the surface, all so curiously related to the twenty-eight bitangents of the general plane quartic curve. If ever there was a fascinating snarl of interlaced theories, Solomon’s seal is one. Synthetic and analytic geometry, the Galois theory of equations, the trisection of hyperelliptic functions, the algebra of invariants and covariants, geometric-algebraic algorithms specially devised to render the tangled configurations of Solomon’s seal more intuitive, the theory of finite groups — all w^ere applied during the second half of the nineteenth century by scores of geometers who sought to break the seal. Some of the most ingenious geometers and algebraists in history returned again and again to this highly special topic. The result of their labors is a theory even richer and more elaborately developed than Klein’s (1884) of the icosahedron. Yet it was said by competent geometers in 1945 that a serious student need never have heard of the twenty-seven lines, the forty-five triple tangent planes, and the twenty-eight bitangents in order to be an accomplished and productive geometer; and it was a fact that few in the younger generation of creative CONTRIBUTIONS FROM GEOMETRY 323 geometers had more than a hazy notion that such a thing as tiie Solomon’s seal of the nineteenth century ever existed. Those rvho could recall from personal experience the last glow of living appreciation that lighted this obsolescent master- piece of geometry and others in the same fading tradition looked back with regret on the dying past, and wished that mathe- matical progress were not always so ruthless as it is. They also sympathized with those who still found the modern geometry of the triangle and the circle worth cultivating. For the differ- ence between the geometry of the twenty-seven lines and that of, say, Tucker, Lemoine, and Brocard circles, is one of degree, not of kind. The geometers of the twentieth century long since piously removed all these treasures to the museum of geometry, where the dust of history quickly dimmed their luster. For those who may be interested in the unstable esthetics rather than the vitality of geometry, we cite a concise modern account 1 (exclusive of the connection with hyperclliptic func- tions) of Solomon’s seal. The twenty-seven lines were discovered in 1S49 by Cayley and G. Salmon 2 (1S19-1904, Ireland); the application of transcendental methods originated in Jordan’s work (1S69-70) on groups and algebraic equations. Finally, in the 1870’s L. Cremona (1830-1903), founder of the Italian school of geometers, observed a simple connection between the twenty-one distinct straight lines which lie on a cubic surface with a node and the ‘cat’s cradle’ configuration of fifteen straight lines obtained by joining six points on a conic in all possible ways. The ‘mystic hexagram’ of Pascal and its dual (1806) in C. J. Brianchon’s (1783-1864, French) theorem were thus related to Solomon’s seal; and the seventeenth century met the nineteenth in the simple, uniform deduc- tion of the geometry of the plane configuration from that of a corresponding configuration in space by the method of projection. The technique here had an element of generality that was to prove extremely powerful in the discovery and proof of cor- related theorems by projection from space of a given number of dimensions onto a space of lower dimensions. Before Cremona applied this technique to the complete Pascal hexagon, his countryman G. Veronese had investigated the Pascal configura- tion at great length by the methods of plane geometry, as had also several others, including Steiner, Cayley, Salmon, and Kirkman. All of these men were geometers of great talent; 324 THE DEVELOPMENT OF MATHEMATICS Cremona’s flash of intuition illuminated the massed details of all his predecessors and disclosed their simple connections. That enthusiasm for this highly polished masterwork of classical geometry is by no means extinct is evident from the appearance as late as 1942 of an exhaustive monograph (xi -f 180 pages) by B. Segre (Italian, England) on The nonsingular cubic surface. Solomon’s seal is here displayed in all its “complicated and many-sided symmetry” — in Cayley’s phrase — as never before. The exhaustive enumeration of special configurations provides an unsurpassed training ground or ‘boot camp’ for any who may wish to strengthen their intuition in space of three dimensions. The principle of continuity, ably seconded by the method of degeneration, consistently applied, unifies the multi- tude of details inherent in the twenty-seven lines, giving the luxuriant confusion an elusive coherence which was lacking in earlier attempts to “bind the sweet influences” of the thirty- six possible double sixes (or ‘double sixers,’ as they were once called) into five types of possible real cubic surfaces, containing respectively 27, 15, 7, 3, 3 real lines. A double six is two sextuples of skew lines such that each line of one is skew to precisely one corresponding line of the other. A more modern touch appears in the topology of these five species. Except for one of the three-line surfaces, all are closed, connected manifolds, while the other three-line is two connected pieces, of which only one is ovoid, and the real lines of the surface are on this second piece. The decompositions of the nonovoid piece into generalized polyhedra by the real lines of the surface are painstakingly classified with respect to their number of faces and other char- acteristics suggested by the lines. The nonovoid piece of one three-line surface is homeomorphic to the real projective plane, as also is the other three-line surface. The topological interlude gives way to a more classical theme in space of three dimensions, which analyzes the group in the complex domain of the twenty- seven lines geometrically, either through the intricacies of the thirty-six double sixes, or through the forty triads of com- plementary Steiner sets. A Steiner set of nine lines is three sets of three such that each line of one set is incident with precisely two lines of each other set. The geometrical significance of permutability of operations in the group is rather more com- plicated than its algebraic equivalent. The group is of order 51840. There is an involutorial transformation in the group for each double six; the transformation permutes corresponding CONTRIBUTIONS FROM GEOMETRY 325 lines of the complementary' sets of six of the double six, and leaves each of the remaining fifteen lines invariant. If the double sixes corresponding to two such transformations have four common lines, the transformations are permutable. If the transformations are not permutable, the corresponding double sixes have six common lines, and the remaining twelve lines form a third double six. Although the geometry'- of the situation may' be perspicuous to those gifted with visual imagination, others find the underlying algebraic identities, among even so impressive a number of group operations as 51840, somewhat easier to see through. But this difference is merely one of ac- quired taste or natural capacity', and there is no arguing about it. However, it may' be remembered that some of this scintillating pure geometry was subsequent, not antecedent, to many' a dreary page of laborious algebra. The group of the twenty'- seven lines alone has a somewhat forbidding literature in the tradition of the late nineteenth and early twentieth centuries which but few longer read, much less appreciate. So long as geometry' — of a rather antiquated kind, it may be — can clothe the outcome of intricate calculations in visualizable form, the Solomon’s seal of the nineteenth century will attract its de- votees, and so with other famous classics of the geometric imagination. But in the meantime, the continually advancing front of creative geometry will have moved on to unexplored territory of fresher and perhaps wider interest. The world some- times has sufficient reason to be weary' of the past in mathe- matics as in everything else. What is geometry? In the typical episode of the nineteenth century' just recalled, we see once more the continual progression from the special to the general, in the emergence of widely applicable methods from laboriously' acquired collections of individual theorems, that characterized mathematics since 1800. The methods generalized from Cremona’s have retained their vitality' and interest, although the particular theorems in which they originated may' have lost their attractiveness for a generation trained in newer habits of thought for which those very theorems were partly responsible. So again, in seeking the things that have endured in mathematics, we are led to processes and ways of thinking rather than to their products in any one epoch. Wc shall see also that the conception of geometry' itself changed with time, until 326 THE DEVELOPMENT OF MATHEMATICS what was called geometry in one stage of the development would hardly have been recognized as such at an earlier stage. Attempts to obtain from professional geometers a statement of what geometry is are likely to be only nebulously successful. Beyond agreeing, more or less, on “geometry is the product of a particular way of thinking,” few geometers will commit themselves to anything less hazy. Accepting this for the present — we shall return to it in other connections — we assume that it has a meaning which can be ‘felt,’ if not understood; and we shall describe some of the main contributions of geometric thought to mathematics as a whole. Numerous representative selections might be made; the topics described here were chosen as an irreducible minimum on the advice of men actively engaged (1945) in developing the geometry of the twentieth century. The principal topics are: the vindication of Euclid’s methodology in the creation of non-Euclidean geometries, and the outgrowths of this in the modern abstract or postulational method; differential geometry from Euler, Monge, and Gauss to Riemann and his successors, with its profound influence on the cosmology and mathematical physics of the twentieth century; the principle of duality in projective geometry, and its final elucidation in the inventions (1831) of J. Plucker (1801-1868, German), also this most original geometer’s conception of the dimensionality of a space; Cayley’s reduction (1859) of metric geometry to projective; algebraic geometry, particularly its connection with Cremona (1863) and birational transformations and the analysis of abelian functions; Klein’s program (1872) for the unification of the diverse geome- tries existing in his day, and the supersession of this program after 1916; and finally, the abstract spaces and topology of the twentieth century, w'-hich some believe to be the beginning of a new type of mathematical thinking. Naturally, only a bare hint of so vast a territory can be given in the space at our disposal; and here as elsewhere in this ac- count we shall note only general trends. A history of any one of the topics would fill a book larger than this. The little de- scribed, however, may stimulate some to find out more about the subjects mentioned. Three of the topics are best considered under analysis, where they will be noted. Klein’s program, its successors, and topology are described in the chapter on invari- ance; what little can be said about the connection between algebraic geometry and analysis is deferred to the theory o CONTRIBUTIONS TROM GEOMETRY 327 functions of a complex variable; and the rise of theories of abstract space is followed in a later chapter as a consequence of the trend toward general analysis, first plainly noticeable in 1906, for which mathematical physics was partly responsible. If, after all this, we are still unable to state what geometry is, we shall at least have caught a glimpse of the mathematics created by geometers in the worship of their inexplicit ideal. Euclid cleared of all blemish In 1733 the Jesuit logician and mathematician G. Sacchcri (1667-1733) completed his involuntary masterpiece, Euclidcs ab omni naevo vindicalus, in which he undertook to prove that Euclid’s system of geometry, with its postulate of parallels, is the only one possible in logic and experience. His brilliant failure is one of the most remarkable instances in the history of mathematical thought of the mental inertia induced by an education in obedience and orthodoxy, confirmed in mature life by an excessive reverence for the perishable works of the immortal dead. With two new geometries, each as valid as Euclid’s, in his hand, Sacchcri threw both away because he was willfully determined to continue in the obstinate worship of his idol despite the insistent promptings of his own sane reason. To ‘prove’ Euclid’s parallel postulate, Sacchcri constructed a bircctnngular quadrilateral by drawing two equal perpendiculars AD, BC at the ends A, B of a straight-line segment AB, and on the same side of AB. Joining D, C, he proved easily that the angles ADC, BCD are equal. The parallel postulate is equivalent to the hypothesis that each of ADC, BCD is a right angle. To ‘prove’ the postulate, Sacchcri attempted to show the absurdity of each alternative. If each angle ADC, BCD is acute (‘hypothesis of the acute angle’), it can be proved that the sum of the angles of any triangle is less than two right angles; if each angle is obtuse (‘hypothesis of the obtuse angle’), the same sum is greater than two right angles; if each angle is a right angle, the same sum is equal to two right angles. Determined to establish the third possibility, Sacchcri deduced numerous theorems from each of the first two hypotheses, hoping to reach a contradiction in each instance. He disposed of the hypothesis of the obtuse angle by tacitly assuming that a straight line is necessarily of infinite length. The hypothesis of the acute angle was rejected by an improper use of infinitesimals. 328 THE DEVELOPMENT OF MATHEMATICS Cleansed by faulty reasoning of all blemish, Euclid’s geome- try shone forth to its worshiper as the absolute and eternal truth, the one possible mathematics of space. Saccheri died happy, unaware that he had proved several theorems in two new geometries, each as sound logically as Euclid’s. The devout geometer had unwittingly demonstrated that his unique idol was but one-third of a trinity, coequal with the others but not coeternal; for no geometry is the everlasting truth that Saccheri thought he had proved Euclid’s to be. It seems rather strange that the good geometer should have rejected the hypothesis of the obtuse angle so confidently; but here perhaps the fault was Euclid’s with his meaningless definition of a straight line. With a precise definition 3 of a straight-line segment as the shortest 4 distance between two points, the concept of a geodesic on a surface is almost immediate. The geodesics on a sphere (which is a surface of constant positive curvature) are arcs of great circles, the analogues of the ‘shortest distances’ in a plane. It is just possible, however, that Saccheri’s disciplined education required him to believe in a flat earth. Any geometry constructed on postulates differing in any respect from those of Euclidean geometry is called non-Eucli- dean. Saccheri’s two rejected specimens were the first non- Euclidean geometries in history. Ever since the time of Euclid, geometers had tried to deduce the parallel postulate from the others of Euclid’s system. No useful purpose is served by cataloguing scores of failures to achieve the impossible, although several disclosed interesting equivalents of the doubtful postu- late. A bibliography 5 of non-Euclidean geometry up to 1911 lists about 4,000 titles of books and papers by about 1,350 authors; and since 1911 the subject has expanded enormously. Much of the more recent work was directly inspired by physics, especially general relativity. Of the tentative steps toward a recognized, valid, non-Euclidean geometry between Saccheri’s (1733) and Lobachewsky’s (1826-9), we need recall only two. In 1766, J. H. Lambert (1728-1777, German) noted that the hypothesis of the obtuse angle is realized on a sphere, and remarked that a novel kind of surface would be required to represent the plane geometry corresponding to the hypothesis of the acute angle. Nothing came of this suggestion till 1868, when E. Beltrami (1835-1900, Italian) showed that the surface vaguely conjectured by Lambert is the so-called pseudo-sp iere. This is the surface of constant negative curvature generate CONTRIBUTIONS FROM GEOMETRY 329 by the revolution of a tractrix about its axis; it had been noted by Gauss, but without application to non-Euclidean geometry. But this belongs to the modern development, and we shall note its peculiar significance later. The first indisputable anticipation of non-Euclidean geome- try was by Gauss. As a boy of twelve, Gauss recognized that the parallel postulate presented a real and unsolved problem; but not till he was well past twenty did he begin to suspect that this postulate cannot be deduced from the others of Euclidean geometry. It is not definitely known when Gauss undertook the creation of a consistent geometry without Euclid’s fifth postu- late. It is certain, however, that he was in possession of the main results of hyperbolic geometry (Klein’s designation for the system constructed on the hypothesis of the acute angle) before N. I. Lobachewsky (1793-1856, Russian) published his complete system (1829), and therefore also before J. Bolyai (1802-1860, Hungarian) permitted his to be printed (1833) as an appendix of twenty-six pages in a semiphilosophical two-volume elementary mathematical work ( Tcntamen , etc.) by his father. It used to be claimed on the flimsiest circumstantial evidence that J. Bolyai had been influenced by Gauss. As it is now generally admitted that there are no grounds whatever for this hypothetical action at a very great distance, we shall pass it with the fact that J. Bolyai’s father, W. Bolyai (1775-1856), was a close friend of Gauss during his student days at the university. Gauss never made any public claim for himself as an inventor of non-Euclidean geometry. His anticipations of a part of what Lobachewsky and J. Bolyai accomplished, almost simultaneously and independently, were found in his papers after his death. Although he himself refrained from publishing the revolutionary geometry, Gauss encouraged others to proceed in their efforts to construct a consistent non-Euclidean system. Two of his correspondents made considerable progress: F. K. Schwcikart (17S0-1S59) and F. A. Taurinus (1794—1874), both German. The latter in particular obtained correct and unexpected results (1825-6) in non-Euclidean trigonometry. The earlier date coincides with that at which J. Bolyai is supposed to have con- vinced himself that hyperbolic geometry is consistent; the later with Lobachcwsky’s first paper, unaccountably lost by the Kazan Physico-Mathcmatical Society, on the new (hyper- bolic) geometry. For reasons that arc not exactly clear, Taurinus 330 THE DEVELOPMENT OF MATHEMATICS destroyed all copies of his own work on which he could lay his hands. In the bald historical statement that Lobachewsky in 1826-9 and J. Bolyai in 1833, almost simultaneously and entirely independently, published detailed developments of hyperbolic geometry, we have recalled one of the major revolutions in all thought. To exhibit another comparable to it in far-reaching significance, we have to go back to Copernicus; and even this comparison is inadequate in some respects. For non-Euclidean geometry and abstract algebra were to change the whole outlook on deductive reasoning, and not merely enlarge or modify particular divisions of science and mathematics. To the abstract algebra of the 1830’s and the bold creations of Lobachewsky and Bolyai can be traced directly the current (1945) estimate of mathematics as an arbitrary creation of mathematicians. In precisely the same way that a novelist invents characters, dialogues, and situations of which he is both author and master, the mathematician devises at will the postulates upon which he bases his mathematical systems. Both the novelist and the mathematician may be conditioned by their environments in the choice and treatment of their material; but neither is compelled by any extrahuman, eternal necessity to create certain characters or to invent certain systems. Or, if either is so conditioned, it has not been demonstrated that he is; and to an adult twentieth-century intelligence the multiplication of super- fluous and mystical hypotheses is a pursuit even more futile than It was in the days of Occam. In reporting this estimate of mathematics by informed opinion in 1945, we must also state that it was by no means universal. Many of the older generation still adhered to the Platonic doctrine of mathematical truths. Nor is there any reason to suppose that Plato will not again reign supreme in the minds of mathematicians. Less rational mysticisms than Plato’s have prevailed since the rediscovery of the virtues of blind irrationalism in 1914. But while the majority of mathematicians still believe they can see through an ancient fraud to the nonen- tity behind it, we shall record briefly how the humanization of mathematics came about. The deflation of older beliefs, how- ever, comprises the main contribution of non-Euclidean geometry to mathematical thought as a whole, and also, per- haps, the principal contribution of mathematics to the progress of civilization. For it seems improbable that our credulous CONTRIBUTIONS FROM GEOMETRY 331 race is likely ever to get very far away from brutehood until it has the sense and the courage to discard its baseless super- stitions, of which the absolute truth of mathematics was one. To appreciate fully the next item of more than local signifi- cance, we must describe a geometrical detail in each of four geometries; and we shall select that one, the existence of par- allels, which precipitated an apparently interminable deluge of non-Euclidean geometries after the hyperbolic geometry of Lobachewsky and Bolyai. In 1854, G. F. B. Riemann (1826- 1866, German) invented a ‘spherical’ geometry, in which Sacchcri’s hypothesis of the obtuse angle is realized. The designations ‘hyperbolic’ and ‘elliptic’ refer to Cayley’s ‘Absolute,’ to be noted later; Euclidean geometry is similarly called ‘parabolic.’ If P is any point in the plane determined by P and a straight line L not passing through P, there is, in parabolic geometry, precisely one straight line, L', through P which does not intersect L; L' is the unique parallel to L through P. In hyperbolic geometry, there are two distinct straight lines L r , L" through P, neither of which intersects L; moreover, no straight line through P and lying in the angle between V and L" meets L. Lobachewsky took L', L" as his parallels to L. In both parabolic and hyperbolic geometry two straight lines intersect in one point; Any two geodesics on a sphere (arcs of great circles) intersect in two points, and there arc no parallels. In Riemann’s so-called ‘spherical’ geometry, space is unbounded but finite; every ‘straight line’ (geodesic) is of finite length; and any two straight lines intersect twice, thus negating Euclid’s postulate that two straight lines cannot enclose a space. Riemann’s ‘elliptic’ geometry can be visualized on a hemisphere if, as in his spherical geometry, ‘straight lines’ arc arcs of great circles of the whole sphere, and if the two extremities of an arc are regarded as the same point. Other ‘realizations’ are easily con- structed for all of the four geometries mentioned; our interest here is in the significance, or lack of it, of all such realizations. This marks the discontinuity in geometry as conceived before and after 1899. The three non-Euclidean geometries mentioned above arc usually called classical; ‘Ricmannian geometry,’ as used since 1916, is distinct from all these geometries. For about thirty years after the invention of hyperbolic geometry, few mathematicians paid any attention to it; and none, it appears now, foresaw what the non-Euclidean gcom- CONTRIBUTIONS FROM GEOMETRY 333 imagined space as a matrix of possible relations. Pasch all but eliminated both these conceptions in his restatement of geometry as a hypothctico-dcductive system in the tradition of Peirce. Instead of attempting to state definitions of points, lines, and planes, as Euclid had done, Pasch accepted these as the un- reduced elements of ‘space,’ and from postulated but unanalyzed relations between these atomistic concepts of his system pro- ceeded to deduction. The relations postulated were abstracted from the accepted geometric notions of centuries of working with diagrams. For example, it was explicitly stated as a postulate that two distinct points determine precisely one straight line. Pasch was therefore closer to Leibniz than to Newton; but he seems still to have believed in the existence of a ‘space’ in Newton’s sense. Whether or not he did, his was the first clear-cut presentation after Euclid’s of geometry as an exercise in postulational technique. Pasch went beyond Euclid in that he realized what he was doing, and did it deliberately, whereas Euclid seems to have been guided by visual imagery, and in consequence to have over- looked hidden assumptions. In any event, Pasch profoundly influenced the geometric thought of his contemporaries and successors. His conception of geometry, now an accepted com- monplace, met the usual opposition encountered by anything new and disturbing. Although neither the physical nor the philosophical meaning, if any, of ‘space’ was affected by the completely abstract reformulation of geometry, the novelty affected some — Veronese of Pascal’s hexagram for one — like a shocking blasphemy suddenly shouted in church. They quickly stopped their cars. Even if this new thing were consistent, it was too arid and too barren to be profitable mathematics. Geom- eters were permitted to retain their feeling for the ineffabilitics of the geometric mode of thought; but geometry itself was reduced to logical syntax. From a distance of fifty years it is difficult to sec why anyone got excited. Intuition and strict logical analysis can live in the same science without killing each other; and what one cannot do, the other can. The next man with courage enough to be unpopular was G. Pcano. At first he escaped notice except in Italy. But when (1 SSS) he began attempting to reduce all mathematics to a precise symbolism which left but few loopholes for vagueness, too-siippery intuition, and loose reasoning, he was regarded with •suspicion. With the help of several Italian collaborators, Pcano CONTRIBUTIONS FROM GEOMETRY 335 a small fraction of twentieth-century mathematics. But they were a potent catalyst for that mathematics, and they attracted hundreds of prolific workers. Among other subjects to profit by the revived and refined methodology of Euclid was non-Euclidcan geometry. One of the most curious geometries invented through a deliberate applica- tion of the postulational method was M. W. Dehn’s non- Archimcdcan system, which linked the similar triangles of the ancient Babylonians to the non-Euclidean geometry of the nine- teenth century. Suppressing the axiom of Archimedes, Dehn constructed a geometry in which similar triangles exist, and in which the angle-sum for any triangle is two right angles. But parallels are not unique as in Euclid’s geometry, an infinity being possible. Equally unforeseen consequences followed on applying the postulational method to projective geometry. The American geometers O. Vcblen and W. H. Bussey con- structed (1906) finite projective geometries in which a ‘plane’ contains only a finite number of ‘points’ and ‘lines.’ These finite geometries reduced the debates of the early nineteenth century on the ‘space’ of projective geometry to sequences of empty noises. Vcblen and J. W. Young produced (1907) a set of completely independent postulates (in the sense of E. H. Moore (1862-1932)) for projective geometry w'hich long remained a standard, and which must have convinced the most obstinate that geometry is a formal hypothetico-deductive exercise in logic. Americans probably did more to exploit the postulational technique after Hilbert’s Grundlagcn than their European colleagues, and their analysis was, on the whole, much sharper and clearer. The method was applied to algebra, geometry, arithmetic, topology, and other subjects by dozens of skilled mathematical logicians, among whom E. V. Huntington (1874—) may be specially mentioned for his exhaustive analyses of postulate systems in many fields. After the formal character of mathematics had been admitted by many, intellectual inertia proceeded once more on its time- honored course. If mathematics, and in particular geometry, is an arbitrary' creation of human beings, then surely the like is not so for traditional logic? In some extrahuman sense the logic which had lasted all of twenty-three centuries must be an absolute that not even mathematicians can defy. We shall sec in the proper place that this absolute, too, was abolished, but not till 1920. 336 THE DEVELOPMENT OF MATHEMATICS Returning to Saccheri (1733), we see now that after all he accomplished his purpose. In a sense that was to prove of in- comparably greater significance for the future of mathematics than a proof of the parallel postulate could ever have been Saccheri had cleared Euclid of all blemish. Although his work was ignored and forgotten for over a century after his death Saccheri deserves as much credit as Lobachewsky and Bolyai for having taken the decisive step toward the abolition of mathematical absolutism. The essential part of Euclid’s doctrine, 'strict deduction from explicitly stated assumptions recognized as such, began to come into its own only with the unconscious creation by Saccheri of non-Euclidean geometry. After more than two thousand years of partial understanding, the creative power of Euclid’s method- ology was gradually appreciated; and Saccheri, had he lived till 1899, would have seen a profounder meaning than he intended in the title of his masterpiece, Euclides ab omni naevo vindicates. The import of what evolved from non-Euclidean geometry transcends geometry in epistemological value. The extramathe- matical by-product seems to stand a better chance of enduring than the vast accumulations of technical theorems which delight geometers, and which tomorrow may join the twenty-seven lines on a general cubic surface in the museum of mathematical art. A meaningless controversy A skilled geometer might devote his entire working life to one species of curves, say hypocycloids or bicircular quartics, and find something new and interesting to himself every day. But he would scarcely come upon general principles. If history is any criterion, generalizations in geometry have not been reached by piling theorem on theorem, but by purposeful efforts to slash through jungles of special results, or by equally con- scious attempts to find out why certain techniques furnish geometric theorems easily, while others demand more labor than their output justifies. A case in point is the prolonged struggle of the synthetic geometers during the first half of the nineteenth century to ascertain ‘why’ analytic methods were, apparently, so much more powerful than those of pure geometry. We sna trace the main lines of this fluctuating struggle in the following section. For the moment we consider the outcome, as this P°^ S1 ‘ bly is the item of greatest interest for mathematics as a w ho c. CONTRIBUTIONS FROM GEOMETRY 337 To state the conclusion first, analytic methods are more powerful than those of pure geometry because the experience of more than a century has shown them to be so. No philosophy of ‘space’ and its ‘geometry’ has added anything of significance to this statement of brute fact; although the profane might suggest that as imaginary points, lines, etc., entered geometry through the formalisms of elementary algebra, and not through visual perceptions of diagrams, any attempt to disguise all algebraic concepts in an elaborate diagrammatic terminology could result only in unprofitable artificialities. This, at any rate, was the prevailing opinion among geometers in the 1930’s. They still continued to exercise their esoteric geometric intuition with telling effect, but only a few persisted in trying to modulate every uncouth configuration through delicately cadcnced in- volutions that would have ravished the pure geometric ear of Poncclet. It is conceded by all that the technical vocabulary of imagi- nary points, etc., is of great utility, and serves much the same purpose as the terminology of analysis. The statement that /(.v) is continuous at ,r = a , for example, condenses several assertions into one that can be used as a unit in deduction. It is unneces- sary to recur in each instance to the meaning of continuity, for its simpler implications have been worked out once for all, and can be applied without further thought. Similarly, the statement that a certain curve is a bicircular quartic, say, implies that each of the circular points at infinity is a node on the curve; and the standard elementary properties of nodes and the so-called circular points are applicable as units in investigating the curve. But this admitted utility of a geometrized algebraic vocabu- lary is not what distinguishes the geometers of the nineteenth century from the majority of their successors. The leading syn- thetic geometers of the past attempted to find the circular points, etc., in the ‘space’ of common experience, because they confused ‘physical’ or ‘real’ or ‘a priori’ space — without attempting to explain what they understood by such space — with the abstrac- tions of their algebra and the inadequacies of their diagrams. The conception of geometry as a hypothctico-dcductivc system was about sixty years in the future when (1822) J. V. Poncclet (17SS-1S67, French) published his Trait c dcs prop rictes projec- tors ckr figures. This classic of the synthetic method was largely responsible for one of the most fruitful and, as is now widely believed, least meaningful contro%-ersics in the history of mathc- CONTRIBUTIONS FROM GEOMETRY 3 39 SaratofF on the Volga, thinking out the pure geometry of his Traitr des proprietes projectiles des figures, which he published in 1822 after his return to France. He tells all, or nearly all, about it in his autobiographical introduction. The reservation, if justified, is hardly an argument in support of Poncclet’s main thesis that analytic methods are inferior to synthetic. For G. Darboux (1842-1917), another great French geometer, let the analytic cat out of the synthetic bag before a large and appreciative audience at the St. Louis (U.S.A.) congress in 1904. According to Darboux, “We know, moreover, by the unfortunate publication of the SaratofF notes, that it tvas by the aid of Cartesian analysis that the principles which serve as the base of the Traite des proprietes projectives were first estab- lished.” As this blunt statement of fact seems not to be generally known to geometers, it may be that others besides Darboux liat'e considered it unfortunate. The proponents of analytic methods might consider this awkward disclosure extremely fortunate. But, by whatever means Poncclet first established his general principles, he put new life into a branch of geometry that was almost dead. Observing that certain properties of a plane con- figuration, such as the collincarity of three points in Pascal’s theorem, are unaltered by projection, Poncclet undertook a systematic investigation of such phenomena and defined the 'graphic’ (our ‘projective’) properties of figures to be those which arc independent of the magnitudes (measures) of dis- tances and angles. As we shall sec later, Cayley forced the metric properties banned by Poncclet into a more inclusive projective geometry, in which the imaginarics that plagued Poncclet arc given a place of honor. Poncclet himself (1822), in accordance with his restatement of Kepler’s principle of continuity, intro- duced the line at infinity, and demanded in his plane geometry that every circle cut this line in the same two imaginary' points. All pairs of simultaneous equations representing circles were thus provided with the correct number of common solutions. G. Mange (1746-1S1S, French) had already used pairs of imaginarics to symbolize real spacial relations; but Poncclet was more concerned with attempting to find a ‘real’ justification for imaginarics in geometry. Poncclet’s principle of continuity amounts to the theorem of analysis, that if an analytic identity' in any' finite number of vari- ables holds for ail real values of the variables, it holds also by' analytic continuation for all complex values. The attempt to 340 THE DEVELOPMENT OF MATHEMATICS ‘realize,’ or to disguise, this elementary fact of analysis in a sup- posedly existent space involved Poncelet in a furious controversy with Cauchy. The analyst insisted that the geometer’s reasoning, if not entirely illegitimate, was a needlessly complicated restate- ment of simple algebra; the geometer justified his tactics by proving numerous theorems with much greater ease than seemed possible by analysis. The dispute ended in a draw. But, as will be noted in connection with topology, Poncelet’s intuition out- ran his logic, and his attempted justification of his famous principle rested on a void. Nevertheless, he continued to derive correct geometrical theorems with astonishing facility. The principle of continuity was generalized in 1874-9 by H. Schubert (1848-1911, German), who went far beyond Poncelet’s boldest in his ‘calculus of enumerative geometry.’ Schubert’s ‘principle of the conservation of number,’ as the generalization was called, likewise rested on nothing that could now be recognized as a foundation. It asserted the invariance of the number of solutions of any determinate algebraic problem in any given numbers of variables and parameters under varia- tion of the parameters, or under substitution of special values for them, in such a manner that none become infinite, due account being taken of multiple solutions and solutions at infinity. This somewhat dangerous method was used with brilliant effect by several of the leading geometers of the nine- teenth century, including M. Chasles (1793-1880, French), J. Steiner (1796-1863, Swiss), Cayley, and J. G. Zeuthen (1839- 1920, Danish) the last of whom profited by the more glaring oversights of his predecessors. Opinion on these subtle questions was still divided in 1945, the geometers affirming that their reasoning was sufficiently rigorous, the algebraists dissenting. All of the men just mentioned are conspicuous, and some famous, in the history of mathematics: Steiner, called by his con- temporaries “the greatest [pure] geometer since Apollonius, who could not write till he was fourteen; Chasles, a creative geometer and the judicious historian of geometry, whose Aperpu, historique sur I’origine et le developpement des methodes en geometric (1837) is still a classic of mathematical historiography, Cayley, the all-round mathematician, whose development (1846—) of the theory of algebraic invariants provided geometers with a new outlook on algebraic curves and surfaces; and last, Zeuthen, remembered in geometry and as an historian o mathematics. Leaving the discredited principle of continuity, we pass o CONTRIBUTIONS FROM GLOME TR Y 341 that of duality which, with its generalizations, left as substantial a residue of new and useful methods in geometry, algebra, and analysis as any mathematical invention of the nineteenth century. In its classic form, the principle scents to have been first clearly stated, but not fully understood, in 1825-7, by J. D. Gergonnc (1771-1859, French). Gergonnc noted that if in certain theorems of plane geometry the words ‘point* and ‘straight line’ be interchanged, with corresponding changes for collincarities of points and intersections of lines, etc., inde- pendently provable ‘dual’ propositions result. He inferred that the original in all cases implies the dual, which therefore need not be proved independently. By this ‘principle of duality,’ Brianchon obtained his theorem as the dual of Pascal’s. Ger- gonnc also noted the corresponding principle in space of three dimensions, point and plane being duals, and the straight line self-dual. At one stroke an already vast empire of geometry was doubled in extent; and it was a self-denying geometer indeed who refrained from the practice, which quickly became epidemic, of publishing lengthy parallel columns of dual theorems. Gergonnc, like Poncclct, was a military man. Both laid claim to the principle of duality. Poncclct insisted that the principle was a consequence of the method of poles and polars, which he had used so brilliantly in his own geometry of conics; Gergonnc contended that poles and polars were not the root of the principle. Gergonnc was right, but a conclusive, simple demonstration of this fact was not forthcoming til! Plucker attacked the question algebraically, and gave the usual explana- tion by means of point and line coordinates to be found in most textbooks on projective geometry. But although Gergonnc was right, Poncclct was not entirely wrong. His contention was all but saved by the fortunate circumstance that the order and class of a conic each equal 2. Failing to make Gergonnc withdraw his claim to the prin- ciple by fair means, Poncclct resorted to foul, and succeeded in demonstrating that although a geometer may have been an officer, he is not necessarily a gentleman. The campaign of personal abuse and defamation of character which Poncclct waged against the comparatively obscure Gergonnc makes the Ncwton-I.cibniz controversy look like a love feast in Arcady. Today the law would interfere; but in the heroic age of projective geometry, mathematicians were free to persecute their defence- less enemies like the heroes some of them had been. The next great contribution to all mathematics for which 342 THE DEVELOPMENT OF MATHEMATICS Poncelet’s methods were partly responsible is in Vieta’s tradition of transformation and reduction. If a reversible transformation between the respective systems of coordinates in two spaces or between different coordinate systems in the same space, can be established, theorems in either system can be translated im- mediately into theorems in the other, the correspondence pro- viding the bilingual dictionary. If the restriction that the transformation be reversible is removed, the dictionary reads one way only, say like French into English, and configurations in the first space are mapped onto others in the second, but not vice versa. In either case there may be certain singular loci which must be excluded from the statements of theorems; an example will be given in connection with birational transforma- tions. There is no limit other than practical utility to the trans- formations that might be constructed in accordance with these very general specifications. If easily manipulated transforma- tions which alter the order of a curve (or surface, etc.) can be produced, the geometric gain is obvious. Numerous useful transformations of the kinds described have been constructed since Poncelet, in his method of reciprocal polars, first made a geometric element of one kind (a point) correspond to one of another kind (a line). Possibly the most extensively investigated transformations have been the bira- tional, in which the coordinates for either of two spaces are rationally expressible in terms of those of the other. This allusion must suffice here, as birational transformations are best described in connection with the theory of algebraic functions and Riemann surfaces, and must be deferred to a later chapter. Continuing with our selection of great principles that originated in projective geometry, we note next one of the least expected and most surprisingly simple generalizations in the evolution of mathematics, Pliicker’s theory of spacial dimensionality. We have already seen that Cayley (1843) and Grassman (1844) independently arrived at the notion of n-dimen- sional space, and that the latter defined the 7 z-dimensional manifolds which were to play a capital part in Riemann s geometry. Pliicker in his Analytische-geometrische Entioickeluitgin (1831) observed not only the analytic meaning of duality but also the germ of an idea which was to generalize the duality of Gergonne and Poncelet far beyond the obvious resources of pure geometry. He noted that the general equation o a COX TRI BU T 10 XS FROM GEOMETRY 343 straight line in plane Cartesian geometry contains two variables and two parameters, and that both the variables and the parameters enter the equation linearly. If the roles of the variables and parameters arc interchanged, the equation becomes that of a point. It is to be noticed that the number of parameters is the same, two, in both cases: the plane is a two- dimensional space, or manifold of two dimensions in points and in lines; and we say that the plane contains points and «> 2 lines. These simple observations were the origin of Pluckcr’s vast generalization. If a class of elements is such that a unique element of the class is specified when any particular numerical values arc assigned to each of n numerical-valued parameters, the class is called a number-manifold, or a number-space, of n dimensions, and is said to contain <»" elements. The n parameters written in a prescribed order are called the coordinates of the general element of the class. For example, the general equation of a conic in the plane of Cartesian coordinates contains precisely five parameters; a particular set of values of the parameters specifies a unique conic; the plane is therefore a five-dimensional manifold, or space, when conics arc taken as the basic space- elements, and it contains °o & such elements. If this seems strange at first, it is no more so than the commonplace that the plane is a two-dimensional manifold of points. In Pluckcr's geometry, dimensionality is not an absolute attribute of space, but depends upon the basic elements constituting the space. A Cartesian plane, for example, is three-dimensional in circles. All of this was generalized early in the twentieth century to spaces in which the values of the parameters are not necessarily numbers; the resulting geometries are those of the various abstract spaces intensively studied since Frcchct’s creation (1906) of the first. These will be noted in other connections. Generalizing the classic duality for configurations of points and lines in plane geometry, Piuckcr stated a principle of duality for any two classes of configurations that have equal dimensionalities and arc both linear in the respective coordinates, equal in number, determining the common dimensionality. Each of the classes is a number-space as already defined. The ‘geometry’ of each may be interpreted in many ways. For example, in the three-dimensional space constituted by all circles in a Cartesian plane, an equation between the three coordinates of a circle defines a family of cc 3 circles. We might 344 THE DEVELOPMENT OF MATHEMATICS proceed, as many did, to investigate in detail the properties of such families, defined by equations of degrees 1, 2, ... * Very simple plane representations, easily visualized, of the classical non-Euclidean geometries have been constructed from families of circles. Precisely as in the familiar Cartesian geometry in which points are the basic elements, in Plucker’s geometry we translate the algebra of systems of equations into properties of families of straight lines, conics, cubics, and so on. The duality which Gergonne and others believed to be an absolute attribute of ‘space’ peculiar to the intuitive, diagramed ‘space’ of ele- mentary projective geometry appeared in Plucker’s geometry as a trivial consequence of an unnecessarily restricted way of choosing systems of coordinates. Plucker’s abandonment of the deceptions of visual intuition for the explicitness of algebra and analysis finished something the classical non-Euclidean geometries had only half done. The arbitrary freedom in the mathematical construction of ‘spaces’ and ‘geometries’ at last made it plain that Kant’s a priori space and his whole conception of the nature of mathematics are erroneous. Yet, as late as 1945, students of philosophy were still faithfully mastering Kant’s obsolete ideas under the delusion that they were gaining an insight into mathematics. As Kant appealed to his mathematical misconceptions in the elaboration of his system, it is just possible that some other parts of his philosophy are exactly as valid as his mathematics. Against this it has been contended that Kant’s mathematics remains ‘true’ in a higher realm of ‘truth’ beyond the comprehension of professional mathematicians, whose grudging science leaves them but little time to explore the really important questions of their subject. The difference of opinion may well be left there. The most significant residue of Plucker’s work was the demon- stration once more that geometry as practiced by geometers is an abstract, formal discipline. It should not be necessary to repeat that any experiences which may have suggested a par- ticular set of assumptions for a geometry are irrelevant to the mathematical development. . On the strictly technical side, Plucker elaborated (1865) m great detail the geometry of what we ordinarily picture as a Cartesian three-dimensional space, the point-space of elementary solid geometry and rigid bodies, but with straight lines instea of points as the basic elements. Since the equations of a genera straight line in the Cartesian space involve exactly four m e pendent parameters, Plucker’s ‘line geometry’ is that of a our CONTRIBUTIONS FROM GEOMETRY 345 dimensions! 'pace. Equations between the coordinates represent various families of straight lines; a family of » : straight lines is called a range, one of « 2 , a congruence, and one of s’, a complex; and these species are further classified according to the decrees of algebraic equations in the four line coordinates. The theory then proceeds partly by analogy frith the familiar geome- try bared on point coordinates in point-spaces of two and three dimensions. For example, instead of the cuacric surfaces defined by the general equation of the second degree in point coordinates ( x. y, z), the geometric configurations defined by an equation of the second degree in line coordinates (p, q. r, s) are investigated and classified into types, analogously to the classification of quadrics into pairs of planes, cones, cylinders, ellipsoids, hvpcrboioids, etc. This particular detail is the geometry of the quadratic line complex; the problem of classifi- cation led to much interesting algebra of the type associated with the reduction of matrices to canonical form. In line geome- try, a curve is visualized as an envelope of straight lines, not as a locus of points. The inevitable question, ‘What was the good of it all:/ is unanswerable. But for those who insist upon a scientific or industrial application for all mathematics, it may be recalled that Plucker’ s line geometry found an immediate interpretation in the dynamics of rigid bodies. A four-dimensional geometry of the late 1860’s thus justified its creation and its existence to a generation that believed in machinery with all its heart, with all its mind, and with all the soul it had. Synthesis versus analysis Returning now to 1827, when A. F. Mobius (1790-1860, Ger- man) introduced homogeneous coordinates in his highly original work. Der IcryeerJrische Calcul, we shall follow the struggle of synthetic methods against coordinates until both emerged victorious 5n the 1860’s. Although no sharp line divided the contestants or separated the geometers of different nationalities, the prolific Italian school preferred synthesis after the 1860*s, while the majority of French, German, and British geometers made greater me of analysis. At the beginning of the period, Steiner was the unapproachable champion of synthesis: Plucker, the unrivaled master of analysis. Plucker is usually regarded as the true founder of the method of homogeneous coordinates, in his Ar.alyiiscke-geomeirisckf r.r.tur:cke lunger. 6IS2S, 18:1). which also exploited abridged 346 THE DEVELOPMENT OF MATHEMATICS notation, usually attributed to E. Bobillier (1797-1832, French) and the simple analytic equivalent of Gergonne’s principle of duality. This was followed in 1835 by the System der analytische Geomelrie, in which, incidentally, curves of the third order were completely classified. Attempts by Cramer and others in the eighteenth century to discipline the swarms of curves of the fourth order had failed. In his Theorie der algebraischen Curven (1839), Plucker had a better success. This, however, was totally eclipsed by a discovery of a new kind and of the very first magnitude, ‘ Plucker’ s equations’ connecting the order, the class and the numbers of double points, double tangents, and points of inflection of algebraic plane curves. Cayley pronounced this discovery one of the greatest in the history of geometry. It be- came one of his life-interests to extend Pliicker’s equations to the singularities of skew curves (twisted curves, curves in space) and surfaces. G. Salmon, a fine geometer and algebraist who abandoned mathematics for theology, also made notable con- tributions to this elusive subject. It has extensive ramifications in modern algebra and analysis, the last through the theory of algebraic functions and their integrals. It was in this prolific third decade of the nineteenth century that line coordinates were invented. It is generally agreed that Plucker and Cayley imagined these coordinates independently. Many special surfaces of some interest were exhaustively investi- gated by both point and line coordinates. Two may be recalled as representative of the more interesting specimens collected and anatomized in this detailed sort of work: Kummer’s (1864) quartic surface, which is the so-called singular surface of the quadratic line complex, and which is represented parametrically, as Cayley was the first to discover (1877), by hyperelliptic theta functions; and the wave surface in optics, parametrized by elliptic functions. Through its generalizations to higher space, Kummer’s surface by itself generated an extensive department of geometry. Numerous French, German, Italian, and British geometers elaborated — perhaps overelaborated — this intncate specialty well into the twentieth century. But the general quartic surface in 1945 still presented unsolved problems, and possibly was too complicated for profitable attack by the weap- ons available. Interest in such matters had declined rapidly m the twentieth century, especially after 1920, and they seemed to belong definitely to a glorious but buried past. While Plucker, Cayley, and many others were creating modern analytic geometry with astonishing rapidity, the most CONTRIBUTIONS FROM GEOMETRY 347 ingenious pure geometer since Apollonius was engaged, with spectacular but severely restricted success, in attempting to forge synthetic geometry into an implement of what he hoped would be universal applicability. Steiner’s Systematised Ent- uickelung der Abhangigkeil geomctrischer Gesialtcn von cinander (1832) unified the classical methods of pure projective geometry and applied them with amazing skill to numerous special problems. Incidentally, Steiner proposed several theorems which, presumably, he had discovered by pure geometry, as challenges to be proved by other geometers. The last detail in one of these defied proof by analytic methods till the early 1900’s. The powerful method of inversion is also attributed to Steiner (1824), although others also invented it. It was inde- pendently come upon (1845) through physical considerations by W. Thomson (Lord Kelvin). He and other physicists applied it effectively in its synthetic form to electrostatics, calling it the method of images. Conversely, problems in potential theory can be disguised as exercises in inversion. The trick works both ways because inversion is a conformal transformation. Inversion was one of the first non-linear transformations to be studied deeply in geometry, although quadratic and cubic transformations had been familiar in algebra since the sixteenth century. Special birational quadratic transformations were used by Poncelet (1822), Plucker (1830), Steiner (1832), and systematically (1832) by L. I. Magnus (1790-1861, German). As a detail of historical interest, Magnus (1833) compounded two quadratic transformations to obtain a quartic transforma- tion, by which straight lines correspond to quartic curves. He was thus enabled to read off theorems on quartic curves from their images in straight lines. In somewhat the same direction, E. dc Jonquieres established (1859) a special correspondence between straight lines and curves of order n with a prescribed multiple point of order r. — 1 . Geometers are interested in these historical minutiae because Cremona, who in 1863 set himself the problem of determining all birational transforma- tions of order r. between the points of two planes, apparently overlooked this earlier work, and as late as 1861 believed that if n > 2 no such transformations exist. Geometers point out that had Cremona been conversant with the algebraic notion of closure, as in a group, he would have drawn the correct inference immediately from what was already well known. However, when he realized his oversight, he made rapid progress. What amounts to a capital theorem in the particular birational transformations 348 THE DEVELOPMENT OF MATHEMATICS named after him will appear in another guise when we discuss algebraic functions. It suffices to note here that M. Noether J. Rosanes (German) and W. K. Clifford (1845-1879, English) proved almost simultaneously (1870) that a Cremona trans- formation can be generated by compounding quadratic trans- formations. The prolific Italian school, from Cremona in the nineteenth century to Severi in the twentieth, developed the resulting algebraic geometry mainly by geometric methods; and indeed the corresponding algebra and analysis quickly become unmanageable. The permanent gain from all this somewhat confused development appears to be the methodology of establishing correspondences between classes of different types of geometric configurations. Another extensive division of geometry that developed from the geometry and analysis of the 1820’s-30’s is concerned with the intersections of a variable curve with the curves of a linear series; yet another, with the geometric properties of the inter- sections of two plane curves; and still another, with geometry on curves and on surfaces; and last, the representation of one curve or surface upon another. Parts of these advanced theories belong to algebraic geometry, parts to analysis, the latter through the parametric representation of curves and surfaces by means of certain special functions intensively studied during the nineteenth century. Nothing further can be said here about these highly technical developments; a little more will appear by implication in connection with analysis. But in taking leave of them, we record that literally hundreds of men from the 1860’s to the 1930’s devoted the best years of their working lives to these kinds of geometry. Where so many did work of high quality, it would be invidious to single out individuals. One, however, was outstanding, the fertile and industrious R. F. A. Clebsch (1833-1872, German)). The decisive battle in the' war between the purists and the analysts lasted twelve years', from 1847 to 1860. The first date marks the publication of K. C. G. von Staudt’s (1798-1867, German) Geometrie der Lage; the second, the revised version of this ‘geometry of position’ in the same author’s devastating masterpiece Beitrage zur Geometrie der Lage (1856, I860).. It may be said at once that the uncompromising purist von Staudt drove the enemy frAm the field, but that the analytic geometers retired in good order with all their machinery intact. The victor was left to enjoy the fruits of his barren victory alone. In proving that geometry could, conceivably, get a on ~ CONTRIBUTIONS FROM GEOMETRY 349 without analysis, von Staudt simultaneously demonstrated the utter futility of such a parthenogenetic mode of propagation, should all geometers ever be singular enough to insist upon an exclusive indulgence in unnatural practices. This may not have been what von Staudt intended; it is merely what he accom- plished. If the total exclusion of algebra and analysis from geometry must result in any game as complicated and as artificial as von Staudt’s, then the game is not worth its candle, and geometric purity has cost more than a normal geometer should be willing to pay. None of this detracts from the merits of what von Staudt did. His purification of geometry remains one of the masterpieces of mathematical reasoning. Somebody, no doubt, had to do once for all what von Staudt did, whether it was worth doing or not. Its lasting contribution to mathe- matics is the unintended self-destruction of the ideal of total geometric purity. Observing that cross ratio involves the concept of distance in the line segments from which the ratio is compounded, and remarking that projective geometry professes to be concerned with those geometric properties that are independent of distance and angle, von Staudt proposed to cut the vicious circle by eliminating measures, and therefore numbers, from geometry. The root of the trouble seemed to be that coordinates or their numerical equivalents, presumably extraneous to projective geometry, were subtly implicit in all the classical developments of the subject. The program of von Staudt would reduce number to form, the exact opposite of what Pythagoras proposed and what Kronecker believed he had accomplished. If both von Staudt and Kronecker achieved their aims, number and form may be one. But it seems more plausible that whatever identity, if any, underlies both is merely an irreducible abstract structure of mathematical logic on which both are based. Such speculations as these, however, were far in the future when von Staudt purified geometry. His theory of what he called ‘throws’ gives a purclv projective algorithm for cross ratio and imaginaries. Most remarkably, the algorithm distinguishes between a com- plex number and its conjugate; conjugate imaginaries appear as the double points of an involution on a real straight line. It is interesting to note here a similarity between von Staudt’s mathematical thought and Dedekind’s: faced with a finite problem in arithmetic, Dedekind resorted to infinite classes in his solution: determined to expel imaginaries from geometry, von Staudt replaced them by infinities of real points. 350 THE DEVELOPMENT OF MATHEMATICS It is sometimes asserted that von Staudt was not wholly successful in his attempt to geometrize real and complex num- bers. The abstract geometries of the twentieth century would seem to support this contention. For although it may be possible to geometrize the numbers with which von Staudt was con- cerned, it seems unlikely that any algorithm whatever could reduce the elements of an abstract space to anything either more or less abstract than what they already are. The problem solved by von Staudt, if he did solve it, is of a kind that has been clearly formulated only by the modern postulational method which was not in existence in the 1850’s. Cayley encountered a problem of the same genus as von Staudt’s in his projective theory of metric geometry. This will be described shortly. Cayley’s projective equivalent (1859) of metric distance is based on cross ratio, and therefore involves the very notion of distance which it was designed to eliminate. Cayley himself was aware of this, but he did not attempt to remove the vicious circle. It is probably correct to say that neither the nature of von Staudt’s and Cayley’s problems, nor the logical analysis necessary for satisfactory solutions, was understood before the twentieth century. The struggle between the purists and the analysts, as typified in two of its heroes, illustrates certain general phenomena in the development of mathematical thought of more than geomet- ric interest. Pliicker’s career might form the basis for a study of mental inertia. Steiner’s contemporaries, as already noted, called him “the greatest geometer since Apollonius.” Some even substituted Euclid for Apollonius in their meed of admira- tion for Steiner’s synthetic genius. Pliicker was not called anything much; he was rather ostentatiously ignored by nearly all the elite of geometry. Or at least he personally felt that his fellow geometers were smugly indifferent to his work; and he abandoned mathematics for physics, where he is still remembered. Toward the close of his life, Pliicker emerged into the light again to compose his great treatise on line geometry, Neue Geometrie des Raumes gegrundet auf die Betrachtung der geraden Linie als Raumelemente , published posthumously (1868 -9) under the sympathetic editorship of Klein. Pliicker’s return to geometry was partly occasioned by the warm appreciation of Cayley for his work. Cayley appears to have been the one first-rate mathematician who had an adequate conception of what Pliicker was doing for geometry. Steiners dazz ing brilliance blinded the majority to Pliicker’s incom- CONTRIBUTIONS FROM GEOMETRY 351 parably more massive achievements. Pliicker’s geometry was neither pretty nor — vile but just word — elegant as Steiner’s was. Steiner flaunted his incapacity for analysis, although some of his colleagues insinuated that “the old fox” knew a great deal more than he would admit and, like Poncelet in his funda- mental work, occasionally concealed in synthesis what he had discovered by analysis. But even if this is no more than a malicious canard, Steiner was a contemporary of Apollonius in this thinking. Apollonius would have understood Steiner’s geometry immediately and, with a few days’ practice, might even have beaten his modern rival at the ancient game. But to understand and appreciate what Pliicker was doing, Apollonius would have needed a new brain of a kind they did not produce in ancient Greece. If anyone in the nineteenth century is to be dubbed the greatest geometer since Apollonius, Steiner now seems to be an unlikely candidate for the honor. Yet fashion turned its broad back on Pliicker and favored Steiner with its sweetest, silliest smile. As has happened more than once in the history of mathe- matics, the man with new and fruitful ideas had to die before he might enjoy whatever satisfaction there may be in the esteem of one’s fellow workers. Projective metrics As the last of the major contributions of projective geometry to mathematical thought which we shall describe, we select Cayley’s reduction (1S59) of metric geometry to projective. Cayley gave details only for plane geometry; but with suitable modifications his method can be extended to space of any finite number of dimensions in which a numerical ‘distance function’ is defined for any pair of elements in the space. Abstracting the familiar intuitive properties of the distance between any two identical or distinct points in a plane, geometers lay down the following postulates for the distance, D(p , q), between the elements p, q of any space whose elements are p, q, r, . . . . (1) To any two elements p , q (identical or dis- tinct), there corresponds a unique real number, their distance, D(p, <?)• (2) D{p, p) ~ 0. (3) D{p, q) 0, if p, q arc distinct. (4) D(_p, q) - D(q, />). (5) D(p, q) + D{q, r) ^ D(p, r). The last is called the triangle (or triangular) inequality; it has already been noted in another connection, and will occur again. It was observed in effect by Cayley and E. Laguerre (183-4- 352 THE DEVELOPMENT OF MATHEMATICS 1886, French) independently that these live postulates for distance have a solution D(p, q) in plane geometry other than the usual one giving the distance between two points as a func- tion of their coordinates by means of the Pythagorean theorem With the new definition of distance, and a corresponding one for angle, Cayley converted metric geometry with its usual definitions of distance and angle into a species of projective geometry. In short, he showed that the metric properties of Euclidean space can be reinterpreted as projective properties. Although the details are too technical for brief description, a hint may be given of Cayley’s approach. The quotations are from his sixth memoir on quantics (1859) and his own notes on it in his collected mathematical papers. . . . the theory in effect is, that the metrical properties of a figure are not the properties of the figure considered per se apart from anything else, but its properties when considered in connection with another figure, viz., the conic called the absolute.” “Metrical geometry is thus a part of descriptive [projec- tive] geometry, and descriptive geometry is all geometry, and reciprocally. . . . Regarding Cayley’s ‘all’ we must remember that he was writing in 1859. Cayley at first honored his ‘absolute’ with a capital ‘A,’ a deserved tribute to the magnitude of his inven- tion. But on learning that the Absolute was commonly used by metaphysical theologians to designate a certain extraspacial, extratemporal Entity, Cayley, who was a devout Christian, hastily descended to lower-case ‘a.’ Cayley’s absolute can be imaginary. It may have been the additive property of collinear dis- tances that suggested Cayley’s projective distance and his absolute. For if p, q, r are collinear points, and if the straight- line segments pq , qr, pr are taken with their proper signs accord- ing to the usual rule, then pq -f- qr — pr. This resembles the theorem for the logarithm of a product. In any event, Cayley defined the distance D(p, q) between two points p, q in terms of a logarithm, as follows. The join of p , q cuts a certain fixed conic, Cayley’s ‘absolute,’ in two points p', q'; when p, ?are any fixed points, the four collinear points p , q, p q', taken m a certain order, determine a unique cross ratio; a constant, k, times the logarithm of this cross ratio is Cayley’s definition of D(p, ?)• It is easily seen that this D(p, q ) satisfies the stated postulates for a distance function. Thirteen years after Cayley’s reduction of metric properties to projective by means of his absolute, Klein (1871) notice that the projective definitions of distance and angle provided a CONTRIBUTIONS FROM GEOMETRY 353 simple unification of Euclidean geometry and the classical non-Euclidean geometries. These geometries, Klein showed, differ basically only in their respective distance functions. In Cayley’s definition, the constant k and the conic fixed as the absolute can be so chosen that the respective classical geometries of Lobachewsky and Bolyai, Riemann, and Euclid arc completely specified according as the absolute is real, imaginary, or degenerate. This striking result of Klein’s was a fitting climax to half a century’s striving for clarity in the projective geometry restored to life by Poncelet. Still greater things were to come a year later (1872) in Klein’s famous Erlangcr Programm. This will be noticed in connection with invariance. Klein’s program domi- nated much of geometry for almost half a century. It was super- seded by younger ideas that became popular only with general relativity after 1916, but which had their origin in Riemann’s revolutionary' work of 1854. We shall consider this next. From cartography to cosmology The problem of constructing fiat maps of the earth’s surface was one origin of differential geometry, which may be roughly described as the investigation of properties of curves and sur- faces in the neighborhood of a point. Still roughly, it is required to specify the geometry of a sufficiently small neighborhood with sufficient accuracy, the specification to be valid for the neighbor- hood of any point on the curve or surface investigated. Another origin of this ‘local’ geometry' was the study', in the seventeenth and eighteenth centuries, of tangents, normals, and curvature, the calculus having provided adequate means for a general attack. A third source is evident in the dynamics of the eight- eenth century, particularly in constrained motion, as in the dynamics of a particle restricted to move on a prescribed surface. With problems of these general types occur also their obvious inverses. For example: given a particular formula for the geo- desic distance between any' two neighboring points, to deter- mine the most general surface for which the formula holds; or to classify surfaces with respect to their lines of curvature. M any of these differential problems have immediate generaliza- tions to space of any finite number of dimensions. The resulting theories, as might be anticipated even from these meager hints, are of vast extent and have close connections with differential equations and mathematical physics. Attempting neither a history' nor a catalogue of what has 354 THE DEVELOPMENT OF MATHEMATICS been done since 1700 in differential geometry, we shall select a few typical incidents in the main line of progress, sufficient to connect the physical algebra already discussed with the analysis, differential equations, mechanics, mathematical physics and the non-Riemannian geometries of the twentieth century to be described in subsequent chapters. The increasing attention paid to quadratic differential forms from Gauss (1827) to Riemann (1854), Christoffel (1869), and Lipschitz (1870), then from Ricci (1887) to Einstein and others (1916—), blazes an easily followed trail from the cartography of the earth’s surface to the mapping of a large sector of cosmology on differential geometry. A map is not necessarily a picture on a sheet of paper. The maps of theoretical physics are mathematical descriptions of physical phenomena. Like so much else in modern mathematics, differential geom- etry got its first real start in the analysis of “the myriad-eyed Euler,” who overlooked nothing in the mathematics of his age, totally blind though he was for the last seventeen years of his life. In 1760 he investigated lines of curvature. This work inspired Monge to his own more systematic investigations (1781) in the same direction, and to his general theory of curvature, which he applied (1795) to the central quadrics. Equally signifi- cant for the future of mathematics was Monge’s elucidation of the solutions of partial differential equations by means of his theory of surfaces. The geometric language in which partial differential equations are frequently discussed originated in this early work of Monge. Another of Monge’s inventions, his descriptive geometry, is of less mathematical interest than his analysis of differential equations, but possibly of greater technological importance. Without descriptive geometry of some sort, the engineering sciences of the nineteenth century would have developed much more slowly than they did. Monge’s scheme for representing solids on one plane diagram by means of two projections, a ‘plan and an ‘elevation’ on two planes originally at right angles to each other before being laid fiat, facilitated the visualization of spacial relations, and provided a uniform graphics for solving such problems as determining the curves in which two or more surfaces intersect. Cut-and-try methods might waste a great deal of metal in fitting two pipes of different dimensions at a given angle. This problem is solved with no waste as one of t e earlier exercises in descriptive geometry. Practical mechanics drawing, without which the construction of modern machinery CONTRIBUTIONS FROM GEOMETRY 355 would hardly be feasible, evolved from Monge’s simple scheme. It is seldom pleasant to give the devil his due; but history com- pels us to state that a problem in fortifications was the origin (1763) of descriptive geometry. The French militarists thought so highly of Monge’s invention that they forbade him to pub- lish it, and for about thirty years kept it a secret for their own use. Monge’s account of the subject was first published in 179 5~6. Continuing with what since about 1920 has been called classical differential geometry, we note the Applications de geometric ti de mechanique (1S22) of E. P. C. Dupin (1784— 1S73, French). Dupin’s work was prophetic in several respects. Al- though the indicatrix was not invented by Dupin, he made more effective use than had his predecessors of this suggestive conic in which a plane parallel to, and ‘infinitesimally near to,’ the tangent plane at any point of a surface intersects the sur- face. Analytically, the indicatrix introduces a quadratic differen- tial form into the geometry of certain curves (the asymptotic lines) on a surface. This is analogous to a method of approximation in mathe- matical physics, where the state of a medium in the neighborhood of a point is obtained to a sufficient degree of approximation by neglecting infinitesimals of order higher than the first in the Taylor expansion of the function expressing the exact state of the medium at any point. This procedure is not universal; but where it is applicable, it is one source of linear differential equations in the physical sciences. Geometrically, the indicatrix is useful in the study of two of the most interesting families of curves on surfaces, the asymptotic lines and the lines of curva- ture. Dupin also investigated triply orthogonal families of surfaces, not as a barren exercise in the differential calculus, but because certain instances of such families are of the first impor- tance in potential theory and other departments of mathematical physics. This aspect of differential geometry will be noted in another connection, when we follow the contributions of physics to mathematics, especially in Lame’s conception of coordinates. Another detail of Dupin’s geometry’ was to assume an unfore- seen significance in the lS90’s, when Klein and M. Bocher (IS67-19IS, U.S.A.) observed that the surfaces called cyclides, invented by Dupin, afford a unified geometric background for a wide class of differential equations of scientific importance. A cyclidc is the envelope of a family of spheres tangent to three fixed spheres. Dupin’s geometry was thus one source of much 356 THE DEVELOPMENT OF MATHEMATICS analysis of the nineteenth century. Triply orthogonal systems of surfaces, for instance, were the occasion for one of Darboux’ more famous works, extending to 56 7 pages, which in turn partly inspired G. M. Green (1891-1919, U.S.A.) to a notable simplification (1913) of the general theory as an application of the so-called projective differential geometry of E. J. Wilczynski (1876-1932, U.S.A.). The last is based in part upon a pair of simultaneous partial differential equations of the second order. Green’s 27 pages (1913) incidentally included the meat of Darboux’ 567. Projective differential geometry as practiced in the third decade of the twentieth century offered an interesting example of national preferences in mathematical technique. The two principal schools, the American and the Italian, sought essen- tially the same objectives, but by radically different methods. Each progressed far in its own direction; both were effectively halted, at least temporarily, by obstacles apparently' inseparable from their respective methods. Theoretically adequate for any problem that might arise in the subject, the American method was retarded by wildernesses of unavoidable calculations. A less prosaic but equally' discouraging difficulty, to be described presently', blocked the Italian approach to the generality of a projective differential geometry' of higher space. The American school followed the lead of Wilczynski, who presented his theory', with numerous applications to special problems, in a scries of memoirs, beginning in 1901, and in a treatise (1906) on the general method. Wilczynski had been a pupil of L. Fuchs (1833-1902, German), under whom he ac- quired a mastery of the theory' of differential equations as it was at the close of the nineteenth century'. It was therefore but natural that lie should base his geometry on a complete independent system of invariants and covariants of a sy'stemof one or more linear homogeneous differential equations. A funda- mental set of solutions of the equations uniquely determines the several geometric objects investigated, up to a projective transformation. Under appropriate transformations, of the dependent and independent variables in the differential equa- tions and in the parametric equations of the accompanying geometric objects, the objects and the forms of the differentia equations are invariant, although the coefficients of . the equa- tions will usually' be changed. The covariants basic, for. t e geometry' are functions of the new coefficients, their derivatives, and the new dependent variables, which differ at most by a CONTRIBUTIONS FROM GEOMETRY 357 factor from the same functions of the original variables and coefficients; an invariant is a covariant not containing the dependent variables or their derivatives. The Lie theory of transformation groups (described here in the chapter on in- variance) is the implement of calculation for obtaining the covariants and invariants as necessary preliminaries to the geometry. Probably almost anyone who has ever seriously attempted to solve differential equations by the Lie theory will appreciate the labor inherent in any such heroic project as Wilczynski’s and agree with Galois that, whatever the nature of its unchallenged merits, the theory of groups does not afford a practicable method for solving equations. Galois of course was speaking of algebraic equations, but his opinion, in the judgment of experts in the Lie theory, carries over to differential equations. Beyond a not very advanced stage of complexity, the calculations become prohibitive to even the most persevering obstinacy. The Italian method circumvented the Lie theory. About 1913 the Italian school headed by G. G. Fubini (1879-19-13) approached projective differential geometry through differential forms, arriving at systems of differential equations of the type from which Wilczynski had started. By restricting the analysis to systems in which the coefficients are legitimately specialized, and thereby simplified, by permissible transforma- tions, the basic covariants arc reduced to fairly manageable shape. The method of calculation is the absolute differential calculus, or tensor analysis, of M. M. G. Ricci (1853-1925, Italian), which was noted earlier in connection with the general progress of recent mathematics toward structure. The Ricci calculus, however, originated in the algebra of quadratic differ- ential forms. It was therefore inapplicable to the higher differen- tial forms hinted at in passing by Ricmann in his dissertation (1854) on the hypotheses which underlie geometry. But these forms arc those appropriate for a projective differential geometry of higher space. The Italian method seemed definitely to be inextcnsiblc to a variety of in dimensions in a space of n > 4 dimensions, for 1 < m < n — 1. Nor is there a covariant quadratic form for these cases. It is noted in another connection that the lack of an absolute calculus for differential forms in higher space may be supplied, if, for example, physical specula- tions should render a serious effort to develop such a calculus scientifically profitable. The Ricci calculus did not come into its own geometrically until it was publicized by the relativists, when the geometers adopted and further developed it. The 358 THE DEVELOPMENT OF MATHEMATICS projective differential geometries of the American and Italian schools do not seem to have attracted physicists. These somewhat miscellaneous details have been recalled to underline the estimate of classical differential geometry v,i| c V, was that of a majority of professionals in the 1920’s. Since its inception in the work of Euler and Monge, differentia! geometry had expanded somewhat lawlessly, until by 1903 it embraced a loosely coordinated collection of special problems and incomplete theories, thrown together with no detectable aim and without any clearly defined objective. Such, for exam- ple, was substantially the opinion of Hadamard. In contrast with this disorderly luxuriance, the differential geometry that became popular with the application of Riemannian geometry to physics and cosmology in general relativity was unified ad given definite aims by the absolute differential calculus, or tensor analysis, of Ricci and Levi-Civita. When at last a uni- formity in method was recognized, interest in classical differ- ential geometry all but collapsed. Numerous special result- obtained in the older tradition had long since passed into the general structure of infinitesimal geometry and analysis; but creative work in differential geometry took a new direction. Among other special developments which had seemed promising in the early 1900’s, but which had lost much of their appeal bj the 1920’s, the projective differential geometry of one American school joined the classics which arc respected but seldom culti- vated. The line of descent from the old to the new, as already indicated, was from Gauss, through Ricmann, to the tensor calculus. Gauss (1827) made the first systematic study of quadratic differential forms in his Disquisitioncs gen cralcs circa superf.au curvas , in which the main theme is the curvature of surfaces. Tie forms investigated are in two variables only. With its relevance for the deformation of surfaces and the applicability of one surface on another, Gauss’ theory is a direct descendant oi cartography. This aspect, however, was not that which sug- gested the far-reaching generalization of differential geometry by Ricmann. Geodesy also was one of Gauss’ major interests (1843, 1847) in applied mathematics; and it too is part!) * matter of quadratic differential forms, the line element on a spheroid being the square root of a quadratic differential form in two variables with variable coefficients. Taking the final step in this direction, Ricmann, in one of the most prolific contn u tions ever made to geometry, passed immediately to the gene:<- CONTRIBUTIONS FROM GEOMETRY 359 quadratic differential form in n variables, with variable coeffi- cients, in his vital classic on the foundations of geometry, Ijbcr die llypothesen zvelche der Geometric zu Grundc liegen , 1854. Ricmann’s taste for speculative philosophy has made parts of his great essay needlessly difficult for mere mathematicians. Fortunately for geometry, Ricmann’s mysterious description of a manifold can be ignored; for when he proceeds to mathematics, he actually uses nothing more abstruse than an n-dimensional number-manifold. It would be interesting to know whether Ricmann imagined himself the originator of this notion. But as he seldom mentions other mathematicians in any of his work, even where it is plain that he has profited by their ideas, it is impossible to say how much, if anything, he owed to others. The general manifolds which Riemann attempted to define, but which he did not use, might be interpreted as the abstract spaces of the twentieth century. The mathematics of Riemann’s geometry interweaves two fundamental themes: a generalization of the Pythagorean theorem to any space (number-manifold) of n dimensions; curvature in such spaces. If , .Vn), (-V 1 ~f* dx 1 , . • . , X n -f- dXji) arc the coordinates of neighboring points in the space, and if ds is the infinitesimal distance between these points, it is postulated that ds- — B Zg,-jdx;dxj, in which the double summation refers to i, j t= I, . . . , ii] the £,7 are functions of #j, . . . , x n ; and £,7 — gj;. In laying down this postulate, Ricmann recognized that it gives a sufficient, but not necessary, specification of an elementary distance which is to retain the cardinal properties of a distance function; and he explicitly stated other possibilities. These had not been exploited (at least in print) as late as 1945, although as early as 1924 H. P. Robertson (1903-, U.S.A.) had investigated the analogue, for these possibilities, of the tensor calculus appropriate for the Ricmannian geometry of general relativity. The metric geometry of a particular Ricmannian space is determined by the g i} - occurring in the ds- for the space. Ignoring special cases, Ricmann proceeded at once to his gen- eralized curvature, guided partly by analogy with the Gaussian theory for a two-dimensional space. He then made the remark- able conjecture that his new metrics would reduce questions concerning the material universe and the “binding forces” holding it together to others in pure geometry. Bolder even than Riemann, Clifford confessed his belief 360 THE DEVELOPMENT OF MATHEMATICS (1870) that matter is only a manifestation of curvature in a space-time manifold. This embryonic divination has been acclaimed as an anticipation of Einstein’s (1915-16) relativistic theory of the gravitational field. The actual theory, however bears but slight resemblance to Clifford’s rather detailed creed! As a rule, those mathematical prophets who never descend to particulars make the top scores. Almost anyone can hit the side of a barn at forty yards with a charge of buckshot. The next long stride after Riemann’s toward modern differ- ential geometry was the determination by Christoffel (1869) of necessary and sufficient conditions that a quadratic differential form of the kind in Riemann’s ds 2 be transformable into another by a general functional transformation on the variables. The same problem was also treated by Lipschitz (1870). Christoffel’s solution proved the more useful. In the course of his analysis, Christoffel invented the process named covariant differentiation by Ricci (1887), and used it to derive a sequence of tensors from a given one. Beltrami and others, especially of the Italian school of geometers, used what are essentially tensors; but it remained for Ricci to isolate and perfect the tensor calculus as an inde- pendent algorithm. The algebra of tensors as a generalization of vectors was mentioned in an earlier chapter. The further development of Riemannian geometry will be noted in connection with invari- ance. We may conclude this sketch with a summary indication of the mathematical reason for the scientific utility of tensors. A functional transformation on the variables of a tensor transforms the tensor into another whose components are linear homogeneous functions of the components of the original tensor. A tensor, like an ordinary vector, vanishes if and only if each of its components vanishes. A transformation of the kind stated is, geometrically, a general transformation of coordinates, when the variables are interpreted as coordinates in a space of the appropriate number ojf dimensions. It follows that if a tensor vanishes in one system of coordinates, it vanishes in all; the homogeneity is the decisive factor. This is equivalent to saying that if a system of equations is expressible as the vanishing of a tensor, then the systcpm will be invariant under all transforma- tions of the variables in the system. But this is precisely the condition imposed by one of the postulates of general relativity on a system of equations, if the system is to be an admissible mathematical formulation of an observable sequence of events in physics or cosmology. CHAPTER 16 The Impulse from Science This chapter is introductory to the six following, in which wc shall describe certain typical developments in the evolution of analysis from the seventeenth century to the twentieth. Analysis, perhaps more clearly than algebra or geometry, ex- hibits the constant influence of science on the general develop- ment of mathematics. We saw that the calculus owed as much to kinematics as it did to geometry'. From the death of Newton (1727) to the twentieth century, science continued to stimulate mathematical inventiveness. Of subsequent additions to mathematics originat- ing at least partly in science, the most highly developed are the vast domain of differential equations, the analysis of many special functions arising in potential theory and elsewhere, potential theory itself, the calculus of variations, the theory' of functions of a complex variable, integral equations and functional analysis, statistical analysis, and differential geom- etry. By 1800, the calculus of variations and differential equa- tions had advanced sufficiently to be recognized as autonomous but interdependent departments of mathematics; the statistical method was still an embryonic possibility in the theory of prob- ability: while the theory* of functions of a complex variable had yet to wait a quarter of a century for systematic development by Cauchy, although some of the basic results were implicit in the applied mathematics of Lagrange and others in the eighteenth century. In following the growth of rigor from 1703 to 1900, we noted a constantly sharpening precision of mathematical logic, and saw that attempts to provide a self-consistent foundation for 551 362 THE DEVELOPMENT OF MATHEMATICS analysis led in the early twentieth century to a period of con fusion and a recognition of the necessity for ever more subtle reasoning. Leaving all doubts behind for the present, we nor enter an untroubled region where the end justifies the mean" The end is the increase of scientific knowledge, to which mathe- matics is but one of several means. As in previous accounts ve shall attend only to typical features illustrative of general trends. There are first two matters of possibly wider significance to be noticed: the influence of eighteenth-century mathematics on society; and the response of society, especially after the Napoleonic era, to mathematical research. Mathematics in the Age of Reason The most significant contribution of eighteenth-century mathematics to civilization was a rational outlook on the phys- ical universe, for which dynamical astronomy and analytic mechanics were mainly responsible. The eighteenth century has been called the Age of Reason, also an age of enlightenment, partly because the physical science of that century attained its freedom from theology. In the hundred years from the death of Newton in 1727 to that of Laplace in 1827, dogmatic authority suffered the most devastat- ing of all defeats at the hands of scientific inquiry: indifference. It simply ceased to matter, so far as science was concerned, whether the assertions of the dogmatists were true or whether they were false. At the beginning of the period, it was customary to seek a teleological explanation for the principles of mechanics to accord with the orthodox theology of the time; when Laplace died, all such irrelevancies had been quietly ignored for forty years. Mechanics had at last come of age. Absolute truth, as revealed by science, fled to pure mathematics, where, according to some, it still resides. The French Revolution, beginning in 1789, accompanied the change; and we might be tempted to ascribe the maturing of the exact sciences wholly to that very thoroughgoing upheaval. But the final liberation had occurred in the preceding year with the long-delayed publication of Lagrange’s analytic mechanics. Here, for the first time, a masterpiece of mathematics an science of the very first rank stood erect on its own mathematics and scientific feet without external support. No mysterious spirit of nature was invoked; the work undertook to describe, not to explain, the mechanical behavior of material systems. THE IMPULSE FROM SCIENCE 363 A free translation of a lew sentences will indicate two of the respects in which Lagrange’s mechanics differed radically from its predecessors in both science and mathematics. In his preface, Lagrange writes: I have set myself the problem of reducing this science [mechanics], and the art of rohnng the problems appertaining to it, to general formulas, whose rimplc development gives all the equations necessary for the solution of each problem. . • . No diagrams will be found in this work. The methods which I expound in it demand neither constructions nor geometrical or mechanical reasonings, but solely algebraic [analytic] operations subjected to a uniform and regular procedure. Those who like analysis will be pleased to see mechanics become a new branch of it, and will be obliged to me for having extended its domain. From this it is clear that Lagrange fully realized the signifi- cance of what he had done. The following quotation, typifying the spirit of the entire work, indicates his grasp of the abstract nature of mathematical mechanics: “The second fundamental principle of statics is that of the composition of forces. It is founded on this supposition: ...” Thus the principles of mechanics arc founded on suppositions , that is, on postulates, and arc not eternal truths revealed to a groping mankind by the grudging generosity of some supernatural intelligence. It is mathematical and scientific rationalism like this that validates the claim of the eighteenth century to be called an age of reason. Such clarity of mind as Lagrange’s, however, was the rare exception among mathematicians and scientists in his day and for over a century after his death in 1S13. Lagrange’s most prominent contemporary in the exact sciences, the self-confident Laplace, convinced himself and two generations of eager philoso- phers that the Newtonian mechanics of the heavens was abso- lutely and eternally true; and on this basis he sought to establish the everlasting stability of the solar system. Almost aggressively hostile to the pretensions of the older absolutism, the would-be skeptic Laplace substituted one dog- matic creed for another. It was largely due to the successes of his own celestial mechanics and his widely appreciated popular exposition of the mathematical consequences of Newtonian gravitation that a crude mechanistic philosophy afflicted nearly all physical scientists and many philosophers of the nineteenth century. Eighteenth-century mechanics was also partly responsible for the speed with which machinery overwhelmed civilization in the early nineteenth century. Instead of remaining the private 364 THE DEVELOPMENT OF MATHEMATICS servant of the intelligentsia, the mechanistic philosophy incon- tinently shared its inestimable benefits with the proletariat Hundreds of thousands to whom Lagrange and Laplace might have lectured for years with no transfer of ideas were converted by the dumb, inerring accuracy of their monotonous machines. That a fully developed abstract theory appeared first in modern mathematics from the applied side is less remarkable than it may seem at first sight. Mechanics and mathematical physics generally had no such crushing burden of tradition to throw off as had geometry*. Mathematical mechanics was little more than a century old when Lagrange saw what it was. Only about sixty years before Lagrange published his mechanics, Saccheri’s willful faith in the sanctity of Euclidean geometry had compelled him to ignore the promptings of his own acute reason. Possibly if Archimedes rather than Galileo and Newton had formulated the basic ‘laws’ of dynamics, Lagrange might have hesitated to deflate the foundations of his system to “sup- positions.” But the postulates of mechanics had not had time to fossilize into eternal truths, and Lagrange was not tempted to outrage his reason — at least in mechanics. But in his effort to rigorize the calculus by basing it on Taylor’s expansion, he was as tradition-bound as Saccheri, and possibly for the same reason. The problems of continuity were as old as those of geometry, and the same almost superhuman intransigency rvas demanded to flout tradition and advance in a totally new direction. The eighteenth century may indeed have been the golden age of reason in philosophy and human affairs generally that it is said to have been. The exact sciences, as we have just noted, also submitted to reason in that hard-headed century. But in pure mathematics, there was a marked decline from the standard which the ancient Greeks set themselves. The best that reason could do when confronted with a problem in continuity was Lagrange’s curious attempt to rigorize the calculus. No classic Greek mathematician could have deluded himself so completely as did Lagrange, the greatest mathematician of his age and one of the greatest of all ages. The feeling for sound reasoning in mathematics seemed to have been temporarily lost. Except only -when logical rigor was almost unavoidable to even moderate competence, as in finite algorithms and combinatorial mathematics, the kind of reason- ing that satisfied the leading mathematicians of the so-called age of reason would have shocked Eudoxus and Archimedes, let THE IMPULSE FROM SCIENCE 365 Archimedes, no mere mathematician but a mechanist of the first rank, was Lagrange’s idol. It may be significant that, of all his own great work, Lagrange prized least highly his contributions to the theory of numbers, where without rigid proof for even the seemingly most obvious theorems there is nothing. These had exacted his greatest efiorts, and he doubted whether they had been worth their cost. There is no record of Archimedes’ having esteemed his practical mechanics above his mensuration of the sphere. Social stimuli since the death of Newton The transition from supcrnaturalism to rationalism in the exact sciences did not take place in a social vacuum. Of several hypotheses to account for the worship of mathcmaticizcd reason in the post-Newtonian age, that of economic determinism is the most elastic. In brief, all the work in celestial and analytic mechanics was occasioned by the demands of navigation and ballistics. However, anyone interested may search the technical works of Laplace and Lagrange on mechanics and find no refer- ence to sailing or gunnery. This docs not disprove the thesis that the initial impulse for the mechanics of the eighteenth century may have been the mercantile desirability of a reliable nautical almanac and the military necessity for hitting whatever is aimed at. It merely illustrates the verifiable fact that once a mathematical theory has been initiated, whatever its origin, it proceeds by a sort of intellectual inertia to become abstract with no application in sight. We shall see many instances as we proceed. Potential theory, for example, in spite of its mechanical origin, ceased long ago in those of its divisions that interest professional mathematicians to have any discernible relevance for science or technology. These abstrusities may become practically useful tomorrow; but only should time be reversed and evolution unfold inward will any application of them become their origin. The like holds for the calculus of variations, except that in this instance theology rather than science was the initial source in the eighteenth century. P. L. M. dc Maupertuis (1698- 1759, French) propounded (1747) a somewhat obscure form of the mechanical principle of least action because he credited his parsimonious deity with an aversion to avoidable effort or other waste. This might be construed as thcologic, not economic, determinism unless, as some might insist, the theology was economically determined in the same way as the ballistics. 366 THE DEVELOPMENT OF MATHEMATICS Asserting that “Nature always acts by the shortest path ” Fermat also had gone behind observable phenomena in deriving his optical principle of least time. But Newton framed no super- fluous hypotheses in determining (1687) the surface of revolution offering the least resistance to motion in the direction of the axis through a resisting medium. Newton’s problem, were it proposed for the first time today, might be attributed to eco- nomic determinism, on account of its possible application to the marine torpedoes which had yet to be imagined in Newton’s backward time. The problem, however, has actually been attrib- uted to Newton’s very early advice to a young friend that the latter study ballistics. From these and numerous other examples that might be cited, it seems clear that the truth of a sociological theory when stretched to include all mathematics may occa- sionally vary inversely as its degree of elasticity. The fact seems to be that, if the mathematicians of the eighteenth century were motivated by anything less obvious than the desire to do mathematics and to earn their livings while following their inclinations, they were unaware of it. The condi- tions under which the great creators worked were basically different from those of the nineteenth and twentieth centuries. If an economic motivation is to be found, it might be profitably sought in the domestic and foreign policies of Frederick the Great of Prussia (1712-1786), Catherine the Great of Russia (1729- 1796), the kings of Sardinia, Louis XVI of France (1754-1793), and Napoleon Bonaparte (1769-1821). The obvious demands of civil, naval, and military engineering made the development of mathematics imperative; and these rulers were clear sighted enough to see that the simplest way of getting mathematics out of a mathematician is to pay his living expenses. At various stages of his career, Euler was attached to the courts of Catherine and Frederick; Lagrange was similarly supported through the government- subsidized Turin Academy, a related military school, and later by Frederick, Louis, and Napoleon. Daniel Bernoulli, often called the founder of mathematical physics, was employed by Catherine. Monge and Laplace were employees o successive French governments in various capacities . from military engineering and the training of government engineers to affairs of state, and likewise for Fourier and a number of e> s distinguished mathematicians. But before the Napoleonic era, once these men had advised their employers on the technic^ questions, usually simple, proposed to them, they were free o TIIE IMPULSE FROM SCIENCE 567 spend their working time as they chose. Consequently, an enor- mous amount of mathematics having no detectable application was created. That much of it proved of practical value years or decades later does not alter the fact that its motivation was not economic. All this was printed with the rest largely at public expense in the proceedings of government-subsidized academies. Up to the Napoleonic era, the learned societies were the most important agencies for the publication of research in mathematics. With the eruption of French TLiberty, Equal it}', Fraternity’ in 17S9, a rapid democratization of mathematical research began. Under Napoleon, the leading French mathematicians earned part of their keep by helping to train civil and military engineers at the Ecole Polytechnique. Others at the Ecole Nor- male Supericure taught prospective teachers. Again the major part of the new mathematics produced by the leaders was of no immediate practical value, nor was it undertaken with a view to possible applications. Napoleon no doubt was partly responsible for this liberality. Provided the schools supplied him with a steady flow of competent civil servants and expert engineers to fill the rather frequent vacancies that might be anticipated in a militaristic regime, he was content. Some of the mathematicians compiled excellent texts for the students; others carefully pre- pared their few lectures a week; and nearly all did research in their ample spare time. The situation was not unlike that in a few of the more enlightened European and American universities of the twentieth century. The next and last marked change in the social status of mathematics and mathematicians dates from the decade 1816-26 following the end of the Napoleonic era in ISIS. The universities and technical schools increased rapidly in importance as centers of mathematical research; the learned societies were no longer financially able to cope with the torrent of new mathematics that gushed from a hundred sources; and, most important of all, what was everybody’s business became nobody’s business. Whatever else may be said for democracy, it has consist- ently fostered the individual freedom of mathematicians. No mathematician in a democracy is constrained to create mathe- matics at public expense. All may earn their livings as they please and find what time they can to advance their hobby. Nor are mathematicians as a class debauched by having their researches printed at public expense, although such work can 368 THE DEVELOPMENT OF MATHEMATICS be neither patented nor copyrighted and may be used by an Vo without payment of any kind. In the United States the more liberal universities subsidize the publications of their staffs It is admitted by the majority of educated persons that a technological and scientific civilization without mathematics is an impossibility. Apparently the most efficient way of getting the necessary job done is to leave it to the initiative of indi- viduals on their own time after a more or less exhausting day’s work. Perhaps rather unexpectedly, the result has been a vastly increased output of mathematics since__the close of the Napo- leonic era over all preceding history. The strongest stimuli have been the constantly growing demand for scientific and technological instruction to keep pace •with the rest of modern civilization, and the vast expansion of media for mathematical publication since the German engineer A. L. Crelle (1780-1855) in 1826 subsidized the first high-grade mathematical periodical. In 1940 there were about 280 such periodicals 1 devoted wholly or in part to the publication of re- search in mathematics, and the pressure on existing outlets was steadily increasing. Few if any of these publications would sur- vive for two months if forced to pay their way in a competitire society. They are supported by the whole mathematical frater- nity without distinction as to race, nationality, or creed. The average mathematician subscribes to as many of them as he can afford, even if but very few of the severely technical articles printed in a year are within his comprehension. It might be difficult to account for this curious phenomenon on strictly economic grounds. A majority of the subscribers do no research themselves; so it cannot be polite hints from superiors that they shed luster or notoriety on their employers by getting their names into print that account for the altruistic subscriptions. Nor will a mathematically literate reader scanning the abstracts of current research agree that any considerable percentage of the articles printed by the hundreds every mont were inspired by economic or other practical needs. _ Several thousand 2 periodicals devoted to engineering an other exact sciences take care of the immediately P ractica applications of mathematics. But these are not the journals to which mathematicians subscribe. Many of. these ©or practical journals pay their way in the competitive mar This, possibly, is the nub of the distinction between p and applied mathematics. THE IMPULSE FROM SCIENCE 369 In following the influence of science on mathematics, it should be remembered that mathematics and its applications are different things. A treatise or a monograph on mathematical physics, for example, may be a mass of formulas and equations from beginning to end, and yet make no contribution whatever to mathematics. If the general fact is not ob%'ious, its extreme cases, as in bookkeeping or the calculation of characteristic functions in quantum mechanics, may illuminate the distinction. As a further aid to comprehension, it is a fair guess that out of a hundred thousand persons picked at random on the streets of New York, or Chicago, or London, or Paris, or Moscow, or Tokyo, not one would know the name of the man whom profes- sional mathematicians almost unanimously considered was the foremost member of their guild since about 1912. He died in 1943, inactive; but his fame is secure even if the average man in the street (or in cultured society) is never likely to hear of him. Of the random hundred thousand, many would instantly name a theoretical physicist who deeply resents being called a mathematician. CHAPTER 17 From Mechanics to Generalized Variables Of all the exact sciences, mechanics, the simplest, has prob- ably been the most influential in the development of modern mathematics. The amount of known mathematics applied to a science is no measure of the importance of that science in mathe- matical evolution; it is the new mathematics inspired by a partic- ular science alone that weighs. Thus in the first twenty years (1925-45) of its existence, quantum mechanics used an enormous amount of mathematics, from special functions to modern algebra, but did not suggest any essentially new mathematics. 1 General relativity, on the other hand, drawing less heavily on mathematics, was directly re- sponsible for the direction taken by differential geometry about 1920. This newer geometry might have been developed almost forty years earlier. All the necessary technique was available; but it was not until the successes of relativity showed that Riemannian space and the tensor calculus were of more than mathematical interest that differential geometers noticed what they had been overlooking. Before considering the mechanical origins of certain parts of analysis, we shall give brief summaries of the relevant progress in mechanics in the eighteenth and nineteenth centuries partly responsible for the mathematics. The search for variational principles One purpose of the eighteenth-century mechanists was the invention of principles from which the mechanics of Galileo an 370 MECHANICS TO GENERALIZED VARIABLES 37 1 Newton could be deduced, and the development of mathematics adequate for the deduction. The main mathematical outgrowths were the calculus of variations; a vast theory of differential equations; a heterogeneous collection of special functions; the beginnings of the theory of line, surface, and volume integrals; more than a hint of ^-dimensional space; 5 the origins of potential theory; and certain basic results in what subscaucntly became the theory of functions of a complex variable. The first comprehensive principle of post-Newtonian me- chanics was D’Alembert’s, published in his Traitc de dynamique (1743): the internal actions and reactions of any system of rigid bodies in motion arc in equilibrium. Or, as often expressed: in a dynamical system the reversed effective forces and the im- pressed forces arc in equilibrium. Supplementing Newton’s principles of the conservation of momentum and of the center of mass, Euler and Daniel Ber- noulli (1700-1782) independently stated (1746) the principle of conservation of areas. All of these foreshadowed the concept of invariance. Euler’s Mcchanica, sive viotus scientia analyticc exposita (1736), was a halfway house between the purely geometrical and synthetic mechanics of Newton’s Principia (1687) and La- grange’s Mechaniquc analytique (1788). Euler sought to replace synthetic methods by analysis, and %vas largely successful. Visual geometric intuition, however, was still used, as in the resolution into tangential and normal components in curvilinear motion. Much of the scrappy geometry of curves and surfaces embellishing antiquated texts on the calculus under the com- prehensive rubric ‘geometrical applications’ originated in this way. Possibly mechanics is also partly responsible for classical differential geometry and the intrinsic geometry of curves. Taking a considerable step toward a general method, C. Maclaurin (1698-1746, Scotch), in his Complete system of fluxions , 1742, advanced beyond Euler by using three fixed axes for the resolution of forces. The advantage of Maclaurin’s procedure over Euler’s is comparable to Descartes’ use of one coordinate system to display any number of curves. But the mathematical formulation of each type of problem still required special devices. Introducing his generalized coordinates, Lagrange in 1760 turned away from mere ingenuity, and started toward the general equations of motion on which he based his analytic 372 THE DEVELOPMENT OF MATHEMATICS mechanics of 1788. The equations of motion for a holonomic dynamical system were then obtained in a form adaptable to the special coordinates most convenient for particular problems. The distinction between holonomic and non-holonomic systems may serve to illustrate certain concepts of mechanics which were partly responsible for the calculus of variations in its earlier form. To exhibit these, it will be necessary to use the “arbitrary infinitesimal displacements” in terms of which applied mathema- ticians frequently think. It is not easy to give a mathematically sound treatment of the related variational operator 3 6 and the infinitesimal displacements that lead to useful results before they finally disappear from the calculations; and one rather extreme school advocates abandoning all pretense of deriving the dynamical equations by such means. It would be closer to modern science to state the general equations of motion as postulates, the sole function of these equations being the mathematical statement of dynamical problems, for which the equations themselves are adequate. Their deduction by more or less mystical reasoning dating from ancient Greece and the Middle Ages is of purely historical interest, contributing nothing to understanding or utility. However, as these vestiges of an older mode of thought are still helpful to the majority of applied mathematicians, we shall follow tradition even where it is now asserted by rigorists to be unsound, and by modernists in theoretical physics to be meaningless. 4 The configuration of a dynamical system, regarded as being composed of material particles subject to constraints (as that all the particles move only on given surfaces) and geometrical conditions (as that the distance between any two given points of a rigid body is constant), is specified at time t by n coordinates <7i, qi, q n , where n is finite. If, for example, the Cartesian coordinates of the rth particle at time t are x r , y r , %r, the system is specified by 3m equations = Mqi, . . . ,q n ), y r = gXqu • • • > ?«)> z, = h r (q u . . . , q n ); r = 1, ' Let each of the generalized coordinates qu ••• !?» receive an arbitrary infinitesimal increment; say the increments are Sq i, . . . , 8q n . There is not necessarily a physically possi. e displacement of the system corresponding to 8q i, • • • > oQm 1 there is, the system is called holonomic; if not, non-holonomic. MECHANICS TO GENERALIZED VARIABLES 373 A holonomic system specified by ?i, . . . , q n is said to have n degrees of freedom. Lagrange’s equations for a system with n degrees of freedom, for which a potential function exists, 5 can now be stated. The derivatives of . . . , q„ with respect to t being de- noted by q i, . . . , q n , and the difference T — V between the kinetic energy T and the potential energy V of the system by L, the equations of motion are d_ (0L\ dt \dq r J T *— 1 , « ■ * , L is called the Lagrangian function, or the kinetic potential, of the system. The point of historical interest here is in the ‘small dis- placement’ (oqt, . . . ,8q„)oi(qi, . . . , q n ). By a route which need not be retraced, its interest being mechanical rather than mathematical, (5qi, . . . , 8q r ) descended from the virtual displacements used by Stevinus, Descartes, and others in statics. Virtual displacements appear fully matured in the principle of virtual work, which was one of the clues followed to an analytic mechanics by Euler and Lagrange. An extremely liberal inter- pretation of ancient and medieval mechanical speculations has enabled some scholars to detect elusive hints of virtual work all the way back to the Greek philosophers. Virtual displacements, virtual velocities, and virtual work arc obviously in the general direction of a calculus of variations. The next major advance in analytic mechanics was in the same direction. It finally reduced the mathematics of statics and dynamics to a topic in the classical calculus of variations. The statement of several mechanical theorems of the seven- teenth and eighteenth centuries had suggested to Euler that all natural phenomena present extrema, and that physical princi- ples, including those of mechanics, should be expressible in terms of maxima and minima. For example, Huygens had shown that Fermat's optical principle of least time holds for media whose index of refraction varies continuously from point to point; James and John Bernoulli had found the catenary as the arc of fixed length passing through two fixed points and having the lowest center of gravity; John Bernoulli’s problem (1696) of finding the curve 5 of quickest descent under gravity from one fixed point to another in a vertical plane had been correctly 374 THE DEVELOPMENT OF MATHEMATICS solved by L’Hospital, Leibniz, Newton, James Bernoulli, and John himself; and last, Euler had sought a function whose variation equated to zero would yield the differential equations of dynamics. For a single particle, the ingenious Euler observed that if the velocity v is given as a function of the coordinates of the particle, the desired equations are obtained by minimiz- ing J» ds, where ds is an element of the path in which the particle moves. Otherwise expressed, the equations of motion are found on performing the variation 8 jv ds and equating the result to zero. In this way Euler was led to minimize definite integrals in which the integrand is of the form f(x, y) (1 -j- y' = dy/dx, and the integration is with respect to x. It is to be noted that Euler, guided by intuition, sought minima to express natural ‘laws,’ possibly because he was of the same pietistic cast of mind as Maupertuis. But Jacobi in his lectures on dynamics (edited, 1866), produced an almost trivial mechanical problem in which the action is a maximum. It is therefore customary to speak of stationary values of definite integrals for the expression of physical laws, rather than to prejudge the issue by expecting a least, a definite integral whose variation vanishes being said to represent a stationary value. The vanishing of the variation is insufficient to secure either a maximum or a minimum, although in many physical situations it is otherwise evident that a definite one of these must occur, and it is seldom necessary to proceed further. But modern science does occasionally demand more than shrewd guessing regarding extrema. Thus, in 1939, R. C. Tolman encountered a capital problem in astrophysics for which scientific intuition seemed insufficient, and for which the more refined techniques of the calculus of variations were at least helpful. Euler’s project was completed in 1834-5 by Hamilton, who showed that the dynamical equations are obtainable from a simple stationary principle, which, for a conservative system, is 8 j^ 1 L dt = 0, where to, t\ are the initial and final times for the passage of a dynamical system with the kinetic potential L from one given configuration to another. A verbal equivalent is as follows. Of all possible motions by which a dynamical system may pass in a given time from one given configuration to another, the actual motion will be that for which the average value of the kinetic potential is stationary. The analytic equiva- lent of ‘possible’ is the process of variation, which evolved, at MECHANICS TO GENERALIZED VARIABLES 37 5 least in mechanics, from virtual displacements. Hamilton also gave a variational principle for non-conservative systems. The variational principles of mechanics are far from ex- hausted by those noted. Thus Gauss (1829) reformulated and generalized D’Alembert’s principle in his own of least restraint; and Hamilton’s principle, also that of least action, were extended in the 1890’s to non-holonomic systems. In his Die Prinzipicn dcr Mecha nil: (1894), H. R. Hertz (1857-1894, German) reworked the subject in terms of geometrical imagery. With the develop- ment of metric differential geometry for space of n dimensions in the second half of the nineteenth century, it was apparently inevitable that the stationary principles of dynamics should be rephrased in the language of geodesies. But here, as elsewhere after Lagrange, whatever gain there may have been was scien- tific rather than mathematical, new techniques in geometry and analysis suggesting reformulations of mechanics. Lie’s theory of contact transformations also was elaborated for its mathematical interest long before it was applied (1889) to unify the differential equations of dynamical systems, although the connection be- tween dynamics and contact transformations was implicit in Hamilton’s work of 1834-5. The like appears to be true of Poincare’s integral invariants (1890), also of the topological methods first applied to dynamics by him and since extensively developed by a prolific company of pure mathematicians. 7 In the last instances, however, outstanding problems of dynami- cal astronomy, such as that of three bodies mutually attracting one another according to Newtonian gravitation, 8 were the ultimate source of the mathematics. 3 Enough has been given to suggest that mechanics was an important source of the calculus of variations. Before passing on to a summary account of the development of that branch of analysis, we may glance at the scientific significance of varia- tional (stationary) principles in general. 3 Competent opinion is sharply divided. In the tradition of Maupertuis, Euler, and their predecessors, one side professes to see cosmic profundities in the derivation of Lagrange’s equations from a variational principle. The profundities are no longer theological, as in the eighteenth century, but concern unapprehended necessities of the physical universe. Conse- quently it is claimed that a real but not wholly understood scientific advance is made when the differential equations of a physical theory arc shown to be obtainable from a variational 376 THE DEVELOPMENT OF MATHEMATICS principle, as was done, for example, quite early by Hilbert in general relativity. The other side holds that variational principles are incapable of adding to science anything not already known in a form better adapted to calculation. For this side, any variational principle in science is at most only a concise restatement of more or less ancient history which might become useful, because it is easily remembered, should all working scientists suddenly forget the mathematics they actually use to obtain results that can be checked against observation. This side further asserts that the reformulation of a science in terms of a variational principle is what the average modern pure mathematician does when he attempts to contribute to science, a task for which he is not fitted. Less immoderate observers occupy a middle position, pointing out that if physicists had scrutinized the duality in Hamilton’s optics and dynamics, in which the principles of least time and least action were shown to be interrelated, they might have come upon de Broglie’s waves and Schrodinger’s wave mechanics about ninety years earlier than they did. But this belongs to the elusive metaphysics of might-have-been, and cannot be considered as a promising suggestion for the future. There remain, then, only the two extremes with no tenable ground between them. Their respective creeds reflect those of the corresponding sides in mathematics. The disciples of Maupertuis would favor the vision of mathematics as eternally existing and necessary truth; their opponents would see mathe- matics only as a humanly created language adapted to definite ends prescribed by human beings. It is a matter of individual preference which of these, if either, is considered the worthier. Functions as variables Problems of maxima and minima in the differential calculus seek those values of the independent variables for which a given function of them assumes a greatest or a least value. The varia- bles represent real numbers. In the calculus of variations it is required to determine one or more unknown functions so that a given definite integral involving those functions shall assume greatest or least values. The variables here are functions. As the simplest example, it is required to find the shortest arc joining two fixed points (*!, (x 2 , yi )* All the infinity of arcs y = /(*)> *1 = x = Xl ’ MECHANICS TO GENERALIZED VARIABLES 377 joining the two points satisfy the end conditions y\ ~ f(x i), y, =/(.Vi). The shortest will be that one (or those, should there be several) of this infinity which makes f: (1 + yydx, where y' denotes dy/dx, a minimum. The solution here is intuitively evident, and therefore open to suspicion. It is also ‘obvious* that the plane closed arc of given length enclosing the maximum area is a circle; and like- wise for a sphere as the surface of given area enclosing the maximum volume. But ‘obvious’ as the last two theorems are, the Greek geometers 10 attempted to prove them by elementary geometry. Discounting the legend of Queen Dido and her bull’s hide, 11 we have in the first of these isoperimetric problems of the ancients the earliest concerning maxima and minima that have been rigorously solved only by the calculus of variations. Solutions were delayed till the second half of the nineteenth century. The simplest mechanical problems of the seventeenth and eighteenth centuries involving minima, such as that of the brachistochrone, 10 transcend intuition but are still within range of ingenious geometry. John Bernoulli in 1697 solved the brachistochrone problem elegantly by special devices, using nothing more advanced than an integration. His brother James far surpassed him in the same year with an inelegant but more general method of solution applicable to a wide class of problems. James Bernoulli’s 13 signal merit was his recognition that the problem of selecting from an infinity of curves one having a given maximum or minimum property was of a novel genus, not amenable to the differential calculus and demanding the invention of new methods. This was the mathematical origin of the calculus of variations. The development of the subject is detailed and intricate, especially in the recent period; and we can give only the briefest summary sufficient to indicate the part played by the calculus of variations in the development of modern analysis. Intrin- sically more difficult than some of the other major divisions of analysis, such as the classical theory of functions, the cal- culus of variations has attracted relatively fewer specialists. But those who have made it their chief concern seem to have been embarrassingly prolific. For our purposes here, the points of greatest interest are the early emergence of the new calculus as an independent 378 THE DEVELOPMENT OF MATHEMATICS department of analysis concerned only incidentally with the mechanical and geometrical problems in which it originated and the progression toward a theory of functions of a non- denumerable infinity of variables. The calculus of variations itself is not concerned with the last; but the minimizing arcs of the theory suggest an infinity of variables in two respects. An extremal (a minimizing or maximizing arc) subject to given end conditions is one of an infinity of variable arcs; the arc itself is an infinite set of points. These hints appear to have been partly responsible for the theory of functions of lines (‘func- tionals’) and the geometry of spaces of an infinity of dimensions. There were roughly six stages in the development. The first extended from the last decade of the seventeenth century to about 1740, and is typified by the work of the Bernoullis. The second opened in 1736 with Euler’s 14 differential equation giving a necessary condition for a minimizing curve. In 1744 Euler gave a systematic exposition of his method, with needed revisions. Abandoning Euler’s semi-geometrical attack, Lagrange (1762, 1770) passed to the third stage with an analytic method which furnished the differential equations of the minimizing curves. He introduced the variational operator 6 and developed its algorithm, greatly simplifying and extending most of the work of his predecessors. With Lagrange, the calculus of varia- tions become an autonomous division of analysis. The fourth stage, 1786-1837, began with Legendre, who investigated the second variation of an integral to find criteria for distinguishing between maxima and minima. This was analogous to the use of the second derivative for the like purpose in problems of maxima and minima solvable by the differential calculus. Legendre’s criteria were inconclusive; Jacobi (1837) gave a critical evaluation of Legendre’s analysis, discussing when it would lead to the desired end and when it would not. Jacobi was thus led to his geometrical interpretation of his own criterion in terms of the conjugate point 16 which he defined. For about forty years — a long time in modern mathematics after Jacobi’s advance, there was no significant progress. But analysis in the meantime was undergoing a basic revision. Weierstrass, “the father of modern analysis,” was transforming the mathematics of continuity into a rigid logical system bearing but little resemblance to the intuitive analysis of most of his predecessors. His lectures 16 of 1879 at the University of Berlin MECHANICS TO GENERALIZED VARIABLES 379 on the calculus of variations mark the beginning of the fifth stage. With almost Gaussian indifference to fame, Weierstrass contented himself with lecturing on his revision of the theory; and although his work was not printed in his lifetime, it pro- foundly influenced the entire future development through the research and teaching of his students. Of the latter, one in particular may be mentioned here, O. Bolza (1857-1942, Ger- man) whose lectures over several years at the University of Chicago were responsible for the highly productive American school in the modern calculus of variations. Bolza attended Weierstrass’ lectures of 1879. In addition to rigorizing the entire subject as it ezisted in his time, Weierstrass made extensive additions of his own. To him are due a new sufficiency condition and the first accept- able sufficiency proofs, for which he invented his fields of extremals. For the geometrical interpretation of his analysis he used the parametric equations of curves, with a consequent gain in generality. This step may have been suggested by the like in differentia! geometry, which had been current since Gauss (1827) made extensive use of it in his study of surfaces. It goes back even farther, to Lagrange’s generalized coordinates in dynamics as functions of the time; but Weierstrass was the first to apply it to the calculus of variations. The Wcierstrassian period lasted into the twentieth century. Its standards of rigor persisted as problems of increasing gen- erality were attacked by modern analysis. There were notable applications to differential geometry in the 1890’s, as in the work of G. Darboux on geodesics, later generalized by several other men. The sixth stage dates from 1899-1900, beginning with Hilbert's proof of his differentiability condition for a minimizing arc, assuring in many problems the existence of an extremal, and his exploitation of the invariant integral since named after him. Finally, in 1921-3, L. Toncili, (1885-, Italian) opened a new chapter, proceeding from PlilberTs work to a revision, concerned principally with existence theorems, of the entire calculus of variations. The individuals whose names have been mentioned are not, of course, the only men whose labors have created the calculus of variations; nor arc the few advances noted an adequate measure of the rich complexity of this subtle division of analysis. Scores of men have contributed hundreds of theorems, until here as cbewhere in modern mathematics what was a narrow specialty 380 THE DEVELOPMENT OF MATHEMATICS in the early nineteenth century began in the twentieth to split into still narrower specialties, each with its assiduous corps cf cultivators. Only an expert who has devoted his working life to the subject can take in the •whole of it or estimate the vitality of its several subdivisions. The same is true for any major department of modern mathematics; and it may be taken for granted that any short report on a particular topic can indicate only a few of the salient characteristics. The same general features as in the rest of recent mathe- matics stand out in the development of the calculus of varia- tions, with one possible difference: some of the most difficult problems appeared early and were partly solved by ingenious men who could not possibly have realized how hard the problems were. Otherwise the progress from special problems to others more inclusive or less restricted followed the familiar pattern of generalization with increasing rigor. Instead of problems concerning arcs with fixed end-points, problems with variable end-points were considered, the earliest being James Bernoulli’s (1697) of the curve of quickest descent under gravity from a fixed point to a fixed vertical straight line, a problem with one variable end-point. Generalization in another direction pro- ceeded by modification of the integrand in the definite integral to be minimized. A third type of generalization combined the first two, superimposing generalized end-conditions on the function to be minimized. A far-reaching generalization of this kind was 0. Bolza’s of 1913, which included several famous problems as special cases, among them Lagrange’s of 1770 and A. Mayer’s of 1878. Since about 1920 the greatest activity in this direction has been in the United States; indeed, shortly after 1900 the calculus of variations became a favorite field of research with American mathematicians, of whom G. A. Bliss (1876-) and his numerous pupils, and M. Morse (1892-) were particularly active. Although we cannot discuss special problems, one may be mentioned for its historic interest. J. Plateau’s (1801-1883, Belgian) problem (1873), first proposed by Lagrange, to deter- mine the surface of least area with a given boundary is solved physically by the soap film which spans a wire model of the boundary. A complete mathematical solution was given only w 1931 by J. Douglas (1897-, U.S.A.). In the calculus of variations we have seen the first extensively developed department of analysis in which functions of variable* MECHANICS TO GENERALIZED VARIABLES 381 other than those discussed in the ordinary calculus are con- sidered. This long step forward was to prove of more than local significance. Much of the analysis of the twentieth century is concerned with functions of generalized variables, and with the corresponding abstract spaces created to provide the appro- priate geometrical description of the analysis. Looking back on the analysis of the eighteenth and nineteenth centuries, we observe many trends toward what has been called general analysis. Enough of these generalizations to indicate the develop- ment of some kind of general analysis, and the need for it, will be described in later chapters. CHAPTER 18 From Applications to Abstractions In the progression toward general theories of analysis, the special functions devised in the eighteenth and early nineteenth centuries for the solution of problems in dynamical astronomy and mathematical physics played a dominant part in determin- ing the course of modern analysis. From the historical record, it seems incredible that some of the special functions, for example those of Bessel and E. Mathieu (1835-1890, French), would ever have seriously engaged the attention of mathematicians had it not been for the initial impulse from science. But not all of the most extensively investigated functions can be credited exclusively to scientific necessity. Thus the multiply periodic functions developed inevitably from the straightforward evolu- tion of the integral calculus. A few typical cases 1 will suffice to illustrate the general trends. A central problem of applied mathematics The hardest thing in any applied mathematics is to strip a scientific or technological problem of enough details, and no more, to bring it within the capabilities of skilled mathematicians and still preserve sufficient of the actual problem to make the solution not utterly irrelevant for practical applications. Observation presents us with no motion immune to friction,, and no incompressible fluid; yet the classical hydrodynamics' of incompressible fluids without viscosity has had many applica- tions. The all-important problem of deciding what concepts to be made central in the mathematical description of natural are phenomena is of a like character, and requires the same rare 382 FROM APPLICATIONS TO ABSTRACTIONS 383 combination of scientific insight and mathematical tact for its successful solution. Velocity in kinematics, entropy in thermo- dynamics, also force, action, and energy in dynamics illustrate the point. A more recent instance is correlation in the statistical method. The great mathematicians of the eighteenth century ex- celled in this most difficult field. The distinction between pure and applied mathematicians did not exist, nor was it necessary, when the Bcrnoullis, Euler, d’Alembert, Clairaut, Laplace, Legendre, and Mongc were at their best. It was largely due to their colossal output of both pure and applied mathematics that it became humanly impossible by the middle of the nineteenth century for a man to attain the first rank as a scientist and as a mathematician. As we look back on all this seething activity, we observe the hesitant beginnings of theories which were to occupy thou- sands of industrious mathematicians from the early nineteenth century to well within the twentieth. Following one of these along the clue provided by the Bessel functions, which are among the most useful functions in mathematical physics, we shall be led to a central problem of applied mathematics. This problem generated numerous special functions; and from these in turn some of the major divisions of modern mathematical analysis evolved. Investigating the oscillations of heavy chains, Daniel Ber- noulli* (1700-1782, Swiss) in 1732 encountered the function later called a Bessel coefficient of order zero. Bessel coefficients of order had appeared earlier in a problem of James Bernoulli’s 4 (1654-1705). The vibrations of a stretched membrane led Euler 5 in 1764 to more general Bessel coefficients, and seven years later Lagrange encountered the same functions in elliptic motion. In 1824, the mathematical astronomer F. W. Bessel (1784-1846, German), needing these functions in his investigation of a perturbative function in dynamical astronomy, developed sev- eral of their more useful properties. Thereafter, the Bessel coefficients and their immediate extensions, the Bessel functions, appeared in physical science almost as frequently as the circular functions, and chiefly for the reasons indicated next. What follows is relevant for our entire subsequent discussion of the influence of the physical sciences on mathematics. I he advantages of special coordinate systems adapted to specific problems were familiar to geometers before a similar 384 THE DEVELOPMENT OF MATHEMATICS specialization in applied mathematics was recognized as an ultimate source of the indispensable special functions, such as Bessel’s, of astronomy and physics. In discussing physical situations involving symmetry about a straight line, for example, it is convenient, indeed almost mandatory, to use cylindrical coordinates (r, <$ , z), just as it is to use spherical coordinates (r, 0, <p) where there is symmetry about a point. When Laplace’s equation 0 V 2 « = 0 is transformed from rectangular to cylindrical coordinates the variables are separable, and Bessel’s differential equation drops out as that which r must satisfy. The same equation appears similarly in the transformation to spherical coordinates of the equation 7 \V 2 u = dv/dt, to which Fourier was led in his analysis of heat conduction. A typical problem of great generality connected with this equation may serve to illustrate the central problem of applied mathematics which we have in view, that of boundary-values, in which special functions, such as Bessel’s, are only details of calculation. The typical problem is to find a solution of Fourier’s equa- tion subject to the following conditions. At each point (x, y, z) of the interior of a homogeneous isotropic solid, the temperature v (satisfying the equation) is to be a continuous function of x, y, z, t, having continuous first and second partial derivatives with respect to x, y, z, and having dv/dt continuous. The tem- perature v throughout the body at the initial time t = 0 is to be given by v — fix, y, z), where / is an arbitrary continuous function; and the solution v, obtained as a function of x, y, s, t, must be such that its limit as t approaches zero is f{jx, y, z). It may be assumed that if two bodies of different conductivities are separated by a common boundary, the temperatures of the bodies at any point of the boundary are the same. 8 The problem is easily modified to take account of radiation into a surrounding atmosphere: the loss of heat per unit area of the boundary is to be proportional to the difference in temperature between the surface and the atmosphere, in accordance with an empirically estab- lished law of cooling. Finally, the temperature at any point (x, y, z) of the boundary at time f may be prescribed as a given continuous function F(x, y, z, /). The solution v of Fourier s equation satisfying these conditions is unique. Special problems of this type leading to Bessel functions are the flow of heat m a circular cylinder or in a sphere whose surface is maintained at zero temperature. FROM APPLICATIONS TO ABSTRACTIONS 385 This typical problem is a specimen of boundary-value prob- lems, in which it is required to construct that solution of a given differential equation, ordinary or partial, that fits prescribed initial conditions. If properly posed, the problem has a unique solution; but, as will appear, it is not always obvious that all the conditions of a given situation have been included in the mathematical formulation, or that, if included, they are analyti- cally compatible. The theory of such problems is coextensive with a vast tract of mathematical physics, and has given rise to equally extensive tracts of pure mathematics connected, if at all, only remotely with practical or scientific applications. Many of the classical boundary-value problems in mathe- matical physics lead to analogues of Fourier’s project of expand- ing an ‘arbitrary’ function /(.v) in a trigonometric series in the to .v-interval — tt tom, say /(.*) = -Uj 0 + X cos nx "b s * n nx )> n *- 1 where the coefficients a 0} a n , b n arc to be determined. Under certain restrictions, the coefficients arc given by - f' r f(y) cos my dy, 7rb a = J_ r f(y) sin my dy, (m £ 0). The point to be noted here is that/(x) is expanded in terms of the solutions cos mx, sin mx, of the ordinary differential equation dhi/dx- -f- vru = 0. A central problem of mathematical physics is a generaliza- tion of this: it is required to expand a suitably restricted func- •a lion f(x) in a scries of the form c D + X f n$n(*), where the n — 1 functions <j>n(x) arc solutions of a given ordinary linear differ- ential equation. The possibility of the expansion being assumed, the problem amounts to calculating the coefficients Co, Cj, c«, . . . . The conditions under which the scries converges must then be determined, if the expansion is to be usable. It seems conservative to say that the majority of those special functions which have been most exhaustively investigated since the early eighteenth century entered mathematics in this way through the differentia! equations of astronomy and physics. Although many of them, like the Bessel coefficients, appeared first in a rather haphazard manner in mechanical problems of 386 THE DEVELOPMENT OF MATHEMATICS I the early eighteenth century, their wider significance began to emerge only with the problem of separation of variables in the partial differential equations of potential theory and other departments of mathematical physics. This led directly to the expansion problem just described, and to the modern theory of boundary-value problems which furnishes the desired coeffi- cients and justifies the expansions. This phase of the general development of analysis will recur frequently as we proceed. Once the special functions had fulfilled the more immediate scientific purposes for which they had been invented, they were exploited by numerous analysts whose interests were purely mathematical. Scientific applications were not even remotely envisaged in the continually refined generalizations 9 of the analysis that had sufficed for physical problems. From one point of view, this rapid transition from the immediately applica- ble to the abstract with no application in sight seems only natural and typical of the general progress of mathematics. Admitting that the development is typical, we may nevertheless question its curiously fortuitous character. The Bessel functions may serve once more as an illustration. It has often been said by analysts with a taste for elegance that no mathematician left to his own devices would ever have dreamed of inventing anything so uncouth mathematically as the Bessel] functions; or, if by cha'nce he had imagined such things in a nightmare, he would have done his utmost to forget them on coming to his senses. Such elegancies as these functions may exhibit in the refinements of twentieth-century analysis, as in the theory of various transforms or in applications to the theory of numbers, were unimaginable to the eighteenth-century mathematicians, whose motives in investigating special cases of the functions were wholly scientific or practical. Whether defined by infinite series or by a differential equation, there was nothing about the Bessel coefficients as first presented to suggest that they and their generalizations might repay ex- haustive investigation on their own account. The like holds for many of the other special functions con- ceived in science and born into technology, for example the Mathieu 10 functions, introduced (1868) to analyze the vibra- tions of an elliptic membrane. All the intricate analysis that developed from these scientific origins seems strangely parasitic and accidental to those who believe that mathematics evolves in response to the dictates of an indwelling and eternal necessity. FROM APPLICATIONS TO ABSTRACTIONS 387 To these, some of the most highly prized acquisitions of modern mathematics arc mere by-products of chance. There is, they maintain, neither reason nor necessity in the selection of what particular things arc to be developed; and almost any choice other than that actually made would produce results equally pleasing to a mathematician. So say the practical realists, who also occasionally take mathematicians to task for fleeing to abstractions when seemingly more fertile fields await cultivation. Against this opinion, it is contended that mathematicians as a class prefer the problems of pure mathematics to those of applied because to do so is merely to follow the line of least resistance. Centuries of trial and error have shown in what directions advances may be anticipated for a moderate expendi- ture of thought; and the same process of elimination has simul- taneously suggested the means of progress. This appears to be the basic reason for the phenomenal popularity of abstract algebra, abstract spaces, and general analysis in the twentieth century. Of the endless variations implicit in the syntax of mathematics at any stage of its development, those following most closely what has already been explicated are usually selected for further elaboration. We shall see a striking and historically important instance of this presently, when we consider elliptic functions. Mathematics and scientific intuition It is ‘intuitively evident’ that electricity applied to a bounded conductor will reach a definite and unique distribution when the conductor is fully charged and no more electricity flows onto it. But it is not evident mathematically. Intuition in mathematics frequently acts as a decoy to credulity. It was so in the evolution of the calculus, and it is so here. The physically evident assertion about the conductor may in fact conceal an ineradicable incompatibility. It may be too sweeping a generali- zation from crude observations. The difficulties begin when intuitive notions of a boundary are made precise. Arc Pcano’s area-filling curves, for example, to be admitted as boundaries ? When these and similar conditions have been agreed upon, intuition has departed. Let intuition state offhand what will be the distribution of electricity on a one-sided conductor, or on a body like a cactus pad with spines tapering off exponentially to infinity. But, it may be legitimately objected, neither of these abnormalities ever ap- 388 THE DEFELOPMENT OF MATHEMATICS pears in nature or technology. Granting this, we are left with the severely practical problems of deciding which conductors sub- mit to mathematical analysis and of excluding from our calcula- tions those that do not. Until these are solved with moderate completeness, our electrostatics will be applicable only ‘in gen- eral.’ That is, it will supply only dubious information. A famous crux of mathematical physics shows just how deceptive intuition unrestrained by reason can be. As this Dirichlet’s principle — was of the first importance in the evolu- tion of analysis in the nineteenth century, we shall describe it in some detail. By a semiphysical argument, Gauss in 1840, and W. Thom- son (Lord Kelvin, 1824—1907, Scotch) in 1847, using the cal- culus of variations, believed they had established the existence of a continuous solution V of Laplace’s equation having assigned values on any given closed surface and minimizing the integral 11 ///[©■-©■+©>**■ the integration extending throughout the volume enclosed by the surface. It is intuitively evident from the physical situation of which this is the mathematical abstraction that the required V exists. Following Riemann (1851), we therefore assert that the mathematical existence of V is assured by that of the physical problem, and call this Dirichlet’s principle, although Dirichlet himself was not so rash as to state it. Dirichlet did, however, follow (1856) Gauss and Thomson in assuming the existence of a minimizing V , a much milder assumption than Riemann’s that, because a problem seems to make sensible physics, it must have a mathematical solution. Unfortunately for intuition, the principle in either form is false. Weierstrass in 1870 proved that the required minimum value of V is not attainable within the domain of continuous functions. What seemed intuitively to be a meaningful problem was thus shown to be a disguised incompatibility. The like holds for the corresponding principle in two dimensions instead of three. _ The -principle being fallacious, what is called Dirichlet s problem supplanted it: to find a function F(x, y, z) which, to- gether with its first and second partial derivatives with respect to x, y, z, shall be uniform (single valued) and continuous FROM APPLICATIONS TO ABSTRACTIONS 3S9 throughout a given closed region R, and •which shall take pre- assigned values on the boundary of R. Dirichlct’s principle was responsible for a vast amount of pure mathematics after Riemann’s appeal to it in the two- dimensional ease in his theory (1851) of functions of a complex variable (to be described in a later chapter). As this theory was one of the most extensively cultivated fields of analysis in the latter half of the nineteenth century, it became important to determine restrictions under which Dirichlet’s problem is solvable. The outcome was a large division of the modern theory of the potential. A list of the developers of this highly specialized topic reads like a directory of the leading analysts from Riemann (1826-1866) to Poincare (1854—1912), and down to the present. For a critical account to 1929, we must refer elsewhere 12 because, after all, potential theory is but one department of dozens in modern analysis, and we can attend here only to general move- ments. Our present interest in the subject is incidental: it is a typical example of the physical origin of much pure analysis, and of the necessity for more than acute physical intuition in the correct formulation and solution of important problems in applied mathematics. We note briefly the scientific and historical origins of the theory. Discussing Newtonian gravitation, Lagrange in 1773 (and 1777) observed that the components of attraction at a given point in space, due to a distribution of mass-particles, arc obtainable as the space-derivatives of a certain function of the positions of the particles. Thus Lagrange invented what is now called the potential V for a Newtonian gravitational field due to a discrete distribution of mass-particles. Laplace (1782) showed that for a point in empty space the potential V due to a continuous distribution of matter satisfies V 2 F — 0; and S. D. Poisson (1781-1840, French) derived (1813) the cor- responding equation V 2 F = — 4rrp for points within the attract- ing mass, the density p at an interior point being given as a function of the coordinates. The next long step forward was taken in 1828 by G. Green (1795-IS4I, English), in his fundamental Essay or. ike applica- tion of mathematical analysis to the theories of electricity and magnetism. This contained the extremely useful result known as Green’s theorem for the reduction of certain volume integrals to surface integrals. 390 THE DEVELOPMENT OF MATHEMATICS It may be noted in passing that Stokes’ (actually Kelvin’s) companion “platitude of mathematical analysis” 13 for the reduction of certain surface integrals to line integrals, which also is of constant use in mathematical physics, made its first public appearance as a problem in a Cambridge examination paper of 1854. Whether any of the examinees solved the problem appears not to be known. But it seems likely that if anyone did turn in a solution acceptable to Stokes, he could not satisfy a modern examiner 14 with the same solution. Like Dirichlet’s problem, Stokes’ theorem, its proof, and its generalizations have developed into a thriving industry of modern analysis. A concise report of what has been done on this detail alone would occupy a chapter. Enough has been said to indicate the strictly physical origin of potential theory, in which Dirichlet’s problem is only an incident, although one of the first importance. It may be re- marked, however, that in the interests of historical justice La- place’s prolific equation should be renamed after Lagrange, who used the equation as early as 1760 in his work on hydrodynamics. The partial rehabilitation of Dirichlet’s discredited principle dates from 1899, when Hilbert proved that under suitable restrictions on the region in tvhich V is defined, on V itself, and on the values assumed by V at the boundary of the region, Dirichlet’s problem is rigorously solvable. But it is no longer intuitive in any sense. The unique historical importance of Dirichlet’s problem is that it was the first in potential theory to raise the question of existence. Double periodicity We pass on to the origins of one of the most extensive depart- ments of nineteenth century analysis, in which practical utility quickly gave way to purely mathematical interest. The history of multiple periodicity is a perfect foil to that of the extremely useful Bessel functions. The knowledge that many natural phenomena are periodic in time, or approximately so, is probably as old as the emergence of the human race from brutehood. Day and night, the recur- rence of the seasons, the waxing and waning of the moon, the physiology of the human body, and many other unescap- able facts of daily life must sooner or later have forced the existence of natural periodicity on even the most rudimentary intelligence. FROM APPLICATIONS TO ABSTRACTIONS 391 Philosophical extrapolations of single periodicity preceded mathematical formulations by thousands of years. Long before Greece was civilized, the sublimely imbecilic vision of Plato’s Great Year, revived in Friedrich Nietzsche’s (1844-1900, German) insane dream of an Eternal Recurrence, had evolved from such banal phenomena as the periodicity of the seasons. Fortunately for the sanity of mankind, poetic philosophers have yet to hear of elliptic functions, whose double periodicity leads at once to a two-dimensional Time. In this infinitely ampler time, with its oo : eternity, history repeats itself indefi- nitely in the parallelograms of a skewed chessboard extending to Infinity in all directions. But the ratio of two sides of any lozenge is real if, and only if, the sides are parallel, when the ratio is Unity. 15 The mathematics (as opposed to the mysticism) of periodic- ity originated in 1748 with Euler’s completely correct determi- nation of the values of the circular functions when the argument is increased by integer multiples of a half-period. Euler, inci- dentally, was the first to emancipate the circular functions from slaver}- to diagrams, and to consider them as numerical- valued functions of a numerical variable. The hyperbolic func- tions, with one pure-imaginary period, followed immediately as obvious consequences of Euler’s exponential forms of the circular functions. They are usually ascribed to V. Riccati 16 (1707-1775, Italian), about 1757; their simple theory was developed in detail by J. H. Lambert (1728-1777, German). None of this indispensable work suggested that more general functions having two distinct periods, and including both the circular and the hyperbolic functions as degenerate cases, might exist. Abel’s discovery in 1825 of these doubly periodic, or elliptic, functions, as they are called, is one of the outstanding landmarks in the history of analysis. The elliptic functions are of the first importance historically, not so much on their own account as for what they instigated. Their singularly rich and symmetrical theory became an invaluable testing ground for the vastly more inclusive theory of functions of a complex variable and for its prolific offshoot, the theory of algebraic functions. These will be considered in a later chapter; for the present we arc interested in the genesis of elliptic functions. The unfortunate term ‘elliptic integral,’ for historical reasons only, designates any integral of the form 392 THE DEVELOPMENT OF MATHEMATICS in which R(z) is a polynomial of the third or fourth degree in z and F(z) is rational in z. The rectification of the arc of an ellipse leads to a special integral of this type; hence the name. Of mechanical problems leading to elliptic integrals, the most elementary is that of finding the duration of one complete oscillation of a simple pendulum. The early work on elliptic integrals has long been of only antiquarian interest. A small sample will suffice to indicate its quality. Being unable to evaluate a special elliptic integral appearing in a problem of elasticity (noted in a later chapter), James Bernoulli in 1694 expressed his conviction that the inte- gration was impossible by means of elementary functions. He was right; but a proof of the impossibility lay far beyond his resources. Maclaurin (1724) translated Bernoulli’s problem into a geometrical construction, which would have been an advance had he shown what means were necessary and sufficient to carry it out. The first work to transcend the obvious was that of the Conti di Fagnano (1682-1766, Italian), who in 1716 proved that two arcs of any given ellipse may be determined in an infinity of ways so that their difference is a segment of a straight line. The significance of this is that Fagnano’s methods are suggestive of those used by Euler 17 in his proof (1761) of the addition theorem for elliptic integrals. But Fagnano’s most remarkable achievement was his discovery that a quadrant of a lemniscate can be divided into n equal parts by a Euclidean con- struction, where « is an integer of the form 2 m h, h = 2, 3, 5. As thus stated, the last may give Fagnano slightly more than his due; but he had the substance of it. The next published hint of a general theory behind such constructions occurs in the Disquisitions s arithmeticae (1801, p. 593, Art. 335), where Gauss remarks that his theory of cyclotomy “can be applied to many other transcendental functions [beyond the circular], for example to those which depend on the integral J = ==•” This particular elliptic integral was one of those discussed by Fagnano in the work just cited; its inversion leads to the special case of elliptic functions sometimes called lemniscatic functions. It would be interesting to know whether Gauss was inspired by Fagnano’s work; Euler frequently expressed his admiration for what his most sagacious predecessor in elliptic integrals had done. FROM APPLICATIONS TO ABSTRACTIONS 593 Another early hint of greater things to come appeared 15 in 1771, in the discovery with which J. Landcn (1719-1790, English) succeeded in astonishing himself: “Thus beyond my expectation, I find that the hyperbola may in general be rectified by means of two ellipses.” Landen's ingenious analytic reformu- lation (1775) of his geometrical theorem is recast today in the transformation of the second order (more generally, of order 2 n , n an integer) in elliptic functions. But all of this early work, including much by d’Alembert, was haphazard in comparison with Euler’s systematic attack on elliptic integrals and their geometrical applications. Em- bedded in an enormous mass of hideous formulas and intricate calculations, two items in Euler’s contribution outrank all the rest in historical significance. The first "was the addition theorem (1761) for elliptic integrals, rated by Euler’s contemporaries and immediate successors as the most amazing tour de force of manipulative skill in eighteenth-century analysis. Euler’s second major contribution was of far greater impor- tance both historically and mathematically, as by an almost ludicrous mischance of fate it misdirected progress for all of forty years after his death in 1783. In the introduction to a memoir 19 of 1764, Euler advocated the incorporation of elliptic arcs into analysis on a parity with logarithms and circular arcs. (Note the italicized word.) Abandoning the fruitless efforts of his predecessors and contemporaries to integrate elliptic differ- entials in finite terms by means of functions then known, Euler boldly proposed that elliptic integrals be recognized as primitive new transcendents to be investigated on their own merits. If this is not what he meant, he proceeded in all of his own analysis as if it were. So great was the momentum of Euler’s algoristic ingenuity that before he could realize his initial mistake he was carried completely out of sight of the right turning which he had missed. That he, of all mathematicians, should have gone astray in this particular matter is one of those mysteries in the evolution of mathematics that pass all under- standing. The master who had initiated the modern theory of the circular junctions failed to observe the greater opportunity which his Providence kept crowding on him and which, had he given it even a casual glance, must have appeared to a mathe- matician of his particular quality as the most natural thing in the world. Instead of considering elliptic arcs as the basic new transcendents, and thereby endowing an already overburdened 394 THE DEVELOPMENT OF MATHEMATICS integral calculus with a new wealth of uncouth formulas, Euler might easily have followed the simple lead of trigonometry. His oversight in adopting the elliptic integrals instead of their corresponding inverse functions as the data of his problem led him into a morass of tangled algebra, precisely as if he had at- tempted to develop trigonometry by an exclusive use of the inverse circular functions — his ‘circular arcs’ — sin -1 x , cos -1 x tan -1 x, etc. The far greater complexity of the theory of elliptic over circular arcs bogged him deeper at each step. Realizing that Euler’s heroic explorations in the wilderness of elliptic integrals had not got the undauntable pioneer very far in spite of many treasures found along the way, A. M. Legendre (1752-1833, French), in 1786 set out on his own explorations. For nearly forty years he followed Euler’s trail, systematizing and civilizing as he went. It is at least conceivable that uncritical reverence for the works of his great predecessor was partly responsible for Legendre’s personal misadventure. More systematic than Euler, and taking more time to his work, Legendre reduced his chaos of refractory material to as coherent a whole as seems to be possible. To him are due the three standard forms of elliptic integrals to which any elliptic integral is reducible. Legendre’s integrals are of course not the only canonical forms possible, and many others have been pro- posed; but Legendre’s retain their usefulness. Forty years of unremitting labor by a master could not fail to produce much of value, if only for its suggestiveness. In particular, Legendre’s work on the algebraic transformation of elliptic integrals directly inspired Jacobi’s first notable success. Legendre presented systematic accounts of his theory in 1811-17, in his Exercices de calcul integral sur divers ordres de transcendentes et sur les quadratures, amplified in 1825-32 in the three volumes (with supplements), Traite des fonctions elliptiques ct des inlegrales euleriennes. The title of the second is responsible for a prevalent confusion in some historical accounts: Legendre s personal work is concerned with elliptic integrals, not with elliptic functions. The distinction, which became of epochal sig- nificance in 1827 with the publication of Abel’s inversion of elliptic integrals, is comparable to that between night and day. Before Abel, nothing was publicly known of elliptic functions as they did not exist outside the private papers of Gauss unti Abel invented them. FROM APPLICATIONS TO ABSTRACTIONS 395 In addition to providing invaluable hints to Abel and Jacobi for the theory of elliptic functions, Legendre’s treatises furnished Cauchy and others with numerous definite integrals, explicitly evaluated, on which to test the efficiency of integration by Cauchy’s method of residues. The like holds for Legendre’s systematization of the beta and gamma functions as they existed in his day. But here again it must be remembered that Legendre’s analysis of 1827 became hopelessly archaic with the creation of modern methods by the great analysts of the nine- teenth century, beginning with Cauchy in 1825. The contrast between the old and the new is strikingly evident on comparing Legendre’s discussion of the gamma function with that of Weicrstrass in 1856, only twenty-three years after Legendre’s death. In taking leave of this fine mathematician of the eighteenth century, we may remember him as a man of the highest char- acter, whose only ambition was the advancement of mathe- matics. If Legendre was so far outdistanced in his own lifetime by younger men — Gauss in arithmetic and the method of least squares, Abel and Jacobi in elliptic functions — it was partly because his own labors had laid the necessary steppingstones. And although Legendre misjudged Gauss and hated him with a venomous hatred, he was the first to welcome and publicize the works of Abel and Jacobi which rendered obsolete his own efforts of forty years. The veteran of seventy-odd not only showed himself incapable of jealousy for his vigorous young rivals in their early twenties, but took pains to understand their work and to expound it in an amplified edition of his own. Such liberality of spirit is no commoner in mathematics than it is elsewhere. Abel revolutionized the subject, and at the same time opened the floodgates of nineteenth-century analysis, in 1827 with a simple remark, “I propose to consider the inverse func- tions.” Instead of regarding the elliptic integral p dx “ ' J V(T-~ckri)(l -f dr) as the primary object of investigation in which a is considered as a function a(.x) of .v, Abel reversed the problem and regarded •v as a function, which he denoted by of a. This inversion of the integral was the essential first step which Abel’s pred- 396 THE DEVELOPMENT OF MATHEMATICS ecessors had overlooked. Its ‘naturalness’ after it had been taken was obvious from the analogy with (* properly restricted) Abel’s first capital discovery 20 concerning the new functions was their double periodicity: <£(# + pi) = 4>(x), 4>(x -f pi) = <j>(x) where pi, pi arc constants whose ratio is not a real number. Thus the elliptic function </>(#) is doubly periodic . Impressed by the great wealth of new ideas that entered mathematics as a direct consequence of Abel’s simple remark, Jacobi some years after Abel’s death characterized inversion as the secret of progress in mathematics: “You must always invert.” If science or mathematics presents us with an awkward situation in which y is given as a function of x, say y = /(*), we should examine the inverse situation, x = / -1 (y), as Abel did when he inverted elliptic integrals and discovered that the inverse functions — the elliptic — are doubly periodic. Jacobi, who balanced his enthusiasm for mathematics with a sense of the ridiculous and who kept his tongue in his cheek when he pontificated, did not intend his prescription to be gulped down as a panacea. He was one of the least professorial of professors who ever lectured to an advanced class. Jacobi’s classic Fundamenla nova theoriac functionum cllipti- carum , published in 1829, the year of Abel’s death, exploited the consequences of inversion and double periodicity, and made the new functions easily accessible to the mathematical public. Even if, as is now generally conceded, Abel’s was the priority in the two basic discoveries, Jacobi made the theory his own and contributed enough to entitle him to rank with Abel as one of its creators. In awarding priority to Abel at the expense of Gauss, we have followed the modern custom of dating ownership from first publication. But with the printing of Gauss’ posthumous papers and the scientific diary which he kept as a young man, it is known that Gauss was in possession of the double periodicity of the lemniscatic function in 1797. Early in 1800 he had dis- covered the general doubly periodic functions, anticipating Abel by a quarter of a century. His posthumous papers also contain numerous formulas relating to the elliptic theta con- stants, rediscovered and brilliantly applied by Jacobi. But Gauss, possibly for lack of an opportunity to develop an PROM APPLICATIONS TO ABSTRACTIONS 397 systematize his discoveries, published nothing on elliptic func- tions. Nor did he make any public claim 21 to have anticipated Abel and Jacobi. In estimating the place of Gauss in mathe- matics, it is customary to credit him with what he actually did. Thus non-Euclidcan geometry and elliptic functions are two of the items which have counted in ranking Gauss with Archimedes and Newton, although he published nothing on cither. Elliptic functions have been given more space than their position relative to modern analysis might justify in a general account, because they clearly mark the beginning of a prolific epoch and were responsible for several major activities in the algebra, arithmetic, and analysis of the nineteenth century. Double periodicity not only opened up boundless new terri- tories; it also marked the definite end of a road which had been followed since Euler’s creation of analytic trigonometry. Jacobi proved (1834) that, if a single-valued function of one variable is doubly periodic, the ratio of the periods cannot be a real number; and that single-valued functions of one variable having more than two periods are impossible. 22 Further technicalities would take us too far off the main road. However, three details of Abel’s and Jacobi’s early work were to prove so prolific of new mathematics all through the nineteenth century that they must be mentioned in passing. The first is Abel’s discovery of complex multiplication, most conveniently described in terms of the Wcierstrassian elliptic function p(«) 5 Esp(ttj«i, w : ), with periods 2«i, 2a?j, arising from the inversion of a certain standard elliptic integral involving the square root of a polynomial of degree three. The choice of p(») implies no restriction. If n is a rational integer, p(nu) is expressible as a rational function of p («). Seeking all other n’s for which a similar theorem holds, Abel found the following unexpected 25 result. If c is a complex number such that p(ca|«i, ut) is rationally expressible in terms of p(«Jo>j, w : ), p(«) is said to admit a complex multiplication by c. In order that such a c may exist, it is necessary and sufficient that Wi/oj; be a root of an irreducible algebraic equation of the second degree with rational integer coefficients. This should be enough to suggest that the theory of complex multiplication is intimately connected with the arithmetic of binary quadratic forms. 24 The develop- ment of this hint occupied scores of algebraists, beginning with Kroncckcr (1857) and Iicrmitc (1859). 398 THE DEVELOPMENT OF MATHEMATICS The second item is Jacobi’s representation of his doubly periodic functions as quotients of what are now called elliptic theta functions. 25 The thetas are not doubly periodic; one of Jacobi’s four is # 3 (*|r) = ^ q ni cos 2mrx, where q ~ e irt r s co 2 /coi, \q\ < 1. The others are obtainable from this by simple linear transformations on x , for example, &s(x + ijr) s t? 4 (*|t). As the values of x for which the thetas vanish are readily deter- mined, the analytic character 25 of the elliptic functions is put in evidence, and from this the Fourier expansions are obtained. The corresponding theta constants ( x — 0) had been investi- gated by Euler in the 1750’s and by Gauss 27 about 1800. But it was only when Jacobi discovered their connection with elliptic functions that their symmetrical theory emerged. Apart from their own extensive theory, the elliptic theta functions proved of great importance as clues to more general theta functions. These will be noted when we come to functions of a complex variable. The third advance that opened up another vast expanse of nineteenth-century analysis also originated with Abel and Jacobi. It is required to exhibit the connections (algebraic relations) between elliptic functions, or between theta functions, whose respective pairs of periods are obtained from one another by linear homogeneous transformations with rational integer coefficients and non-vanishing determinant. This, the trans- formation theory, includes as special cases the problem of real multiplication, as for p(nu ) noted above, and that of division of the periods by a rational integer. As will appear later, a single detail of this theory, that of the elliptic modular functions, expanded in the late nineteenth century into an independent branch of mathematics. Its connection with the general quintic was noted in an earlier chapter. As might be anticipated from its formulation, the general problem draws heavily for its modern solution on the theory of linear groups. Even these meager hints, displayed against a background of unprecedented activity in all departments of mathematics, should suffice to suggest the extent and intricacy of the theories that evolved from Abel’s discovery of double periodicity. Each of half a dozen or more leaders elaborated the entire theory 28 or some favored subdivision according to his persona FROM APPLICATIONS TO ABSTRACTIONS 399 conception of symmetry and grace. Eighty years of this rugged aestheticism endowed analysis with a welter of conflicting notations and trivial distinctions without much of a difference, through which even an expert picks his way with exasperation. Almost in spite of themselves, the leaders rapidly acquired hosts of partisan followers. Mathematicians of all capacities began swarming into the new territory within a decade of its discovery. Several were pupils of Jacobi, but others quickly found leaders with different ideals. The causes of this mass migration are not far to seek. Unlike the special functions devised primarily for the solution of physical problems, such as the Bessel coefficients, the elliptic functions seemed to have been created to round out and extend the integral calculus as it had evolved since the days of Newton and Leibniz. With the applications to the theory’ of numbers by' Kronccker and Hermitc in the late 1850’s, it seemed also as if Gauss must have elaborated his arithmetical theory of binary quadratic forms especially’ for these unforeseen consequences of Abel’s and Jacobi’s early’ discoveries. Comprehensive syntheses to correlate these unexpected coincidences were sought as the century' aged, and were found in the Galois theory of equations, the algebraic theory of fields, and the arithmetic of quadratic number fields. Algebraic curves and surfaces 15 also absorbed enormous quantities of elliptic functions. Applications to classical applied mathematics were made simultaneously, especially to rigid dynamics and problems in potential theory'. But it must be admitted that most of these practical applications have remained of greater interest to pure mathematicians than to working scientists. The rotation of a rigid body', 30 for example, yields numerous elegant exercises in the elliptic theta functions; but few engineers who must busy’ themselves with rotation have time for elegant analysis. The like holds for the occurrence of elliptic functions in practical applications of conformal mapping. When faced with one of these enticing horrors, the experienced designer turns to his drafting board. Contrasted with the Bessel functions, the elliptic functions arc incomparably’ more beautiful and less useful. Yet — or possibly' on that account — they were preferred many’ to one by’ the leading mathematicians of the nineteenth century because, in a sense that any mathematician will un- derstand, they were closer to the ‘natural’ development of mathematics. CHAPTER 19 Differential and Difference Equations Continuing with the mathematics directly inspired by science, we shall indicate next four of the principal stages, not discussed in other connections, by which differential equa- tions became a major discipline of modern pure mathematics. In its later development, this great episode is complementary to the evolution of the Galois theory of algebraic equations and the emergence of algebraic structure. Once more we shall see mere ingenuity being gradually displaced by coordinated attacks, and again we shall note the distinction between mathe- matics as practiced in the recent period and nearly all that preceded the nineteenth century. Discounters of ingenuity do not mean to disparage intuition and insight in any assault on basically new problems. They merely emphasize that the characteristic strategy of modern mathematics favors the mass attack, where feasible, rather than any number of brilliantly executed raids. General methods, not individual gains, are the order of the modern day in mathe- matics. Ingenuity still has its function, even in a general offen- sive; but it is of a more comprehensive kind than any that sufficed in the past. The problems of modern mathematics are not isolated, and to overcome them coordinated efforts on a wide front are increasingly necessary. Five stages The first stage in differential equations, opening with Leibniz in the 1690’s, closed about seventy years later. Roughly, what was accomplished in this period amounts to the first eig 400 DIFFERENTIAL AND DIFFERENCE EQUATIONS 401 weeks’ work in the usual introductory college course. Remember- ing that mechanics, dynamical astronomy, and mathematical physics were intensively cultivated all through and after this period and that numerous problems of analysis originated thus, we must also bear in mind that there was no adequate discussion of differential equations before Cauchy in the 1820’s obtained the first existence theorems. This inaugurated the second stage. The third opened in the 1870’s-80’s with the application by M. S. Lie (1842-1899, Norwegian) of his theory of continuous groups to differential equations, particularly those of Hamilton- Jacobi dynamics. The fourth stage, beginning in the 1880’s with the work of E. Picard (1856-1941, French) developed natu- rally from the third. Here the aim was to construct for linear differential equations an analogue of the Galois theory of algebraic equations. Each stage after the first marked a definite and abrupt advance. The second paralleled the rigorizing of the calculus by Cauchy, and might have been anticipated from the general trend in analysis. The third, Lie’s, even in retrospect, appears to have been unpredictable. Each of the periods has left a sub- stantial residue in living mathematics; the last three posed many problems which still engage scores of specialists. What may be the beginning of a fifth stage opened in the 1930’s, paralleling the modern development of abstract algebra. We pass on to a brief indication of some of the outstanding acquisi- tions in each of these stages and the accompanying developments in finite differences. Before the last three stages can be dis- cussed, the concept of invariance must be described. This will be done in the next chapter. The reign of formalism Both Newton and Leibniz in the seventeenth century solved simple ordinary differential equations of the first order. It seems to have been believed in this earliest stage that the func- tions then known would suffice for the solution of the differential equations arising from problems of geometry and mechanics; and the aim was to find such explicit solutions, or to reduce the solution to a finite number of quadratures. Even when a solution was exhibited as a quadrature, it does not seem to have been suspected that the required integration might necessitate the invention of new transcendents. In fact, it was not until the ISSO's — a stretch of two centuries — that definite knowledge 402 THE DEVELOPMENT OF MATHEMATICS concerning the extreme rarity of differential equations integrate in this rudimentary sense was obtained. Very roughly, if a differential equation is written down at random, the odds against its being solvable in terms of known functions or their integrals are infinite. The first faint hint of generality was Newton’s (1671) clas- sification of ordinary differential equations of the first order into three types, and his method of solution by infinite series. 1 The coefficients of the assumed power-series solution were found as usual. There was no discussion of convergence. Without explicit statement of the assumption, it was assumed that the existence of a physical problem guarantees the existence of a solution of the equivalent differential equation. This seemingly reasonable supposition remained unquestioned in applied mathe- matics from Newton to Riemann. Its viciousness was first unmasked, as we saw in connection with Dirichlet’s principle, only in 1870. Another early forward step was taken by Leibniz, who stumbled on the technique of separating variables. Nearly two centuries were to elapse before Lie’s theory showed when and why this familiar device should succeed. Among other early advances, the homogeneous linear differential equation of the first order was reduced to quadratures by Leibniz (1692); and James Bernoulli (1690) solved the equation of the tautochrone by separation of variables. His brother John (1694) circum- vented dx/x, not well understood at the time, by first applying an integrating factor. Incidentally, the discovery of integrating factors proved almost as troublesome as solving an equation. Another hint of more general tactics appeared in Leibniz’ (1696) change of the dependent variable. John Bernoulli also used this device. By the end of the seventeenth century all the usual elementary and inadequate tricks for first-order equations were known. In addition to problems of the differential calculus on tan- gents, normals, and curvature of the types common as exercises in textbooks, the calculus of variations also had stimulated in- genuity in solving differential equations. Thus James Bernoulli s (1696) isoperimetric problem (noted in another connection) led to a differential equation of the third order which John reduced to one of the second. Before 1700, John also attacked the general linear homo- geneous differential equation with constant coefficients, lo DIFFERENTIAL AND DIFFERENCE EQUATIONS 403 dispose of this detail here, a complete discussion of such equa- tions was given in 1743 by Euler, who also (1741) devised the classical method for non-homogeneous linear equations. The name of Count Riccatr (1676—1754, Italian) is familiar to every student in a first course. What is usually called Riccati’s equation (1723) persistently defied solution in finite form. In accordance with the taste of the age, the ‘real’ problem was to impose sufficient restrictions on the variables to render the transformed equation finitely solvable by separation of varia- bles. The Bcrnoullis claimed to have at least partial solutions; and in 1725 Daniel noted that if m is of the form —4n/(2n ± 1), where it is a positive integer, dyjdx -f- ay- — bx n is solvable in finite terms. By 1723 at latest, then, it was recognized that even an ordinary differential equation of the first order docs not necessarily have a solution finitely expressible in terms of elementary functions. But anything approaching a proof of the impossibility, in general, of such a solution lay far in the future. Singular solutions were noted unexpectedly early, the first 3 instance being due to Taylor (of Taylor’s series) in 1715. Clair- aut, whose name decorates a special type of equation in a first course, followed in 1734 with